No ascetic discipline is so universally recommended by Scripture and Tradition as the practice of fasting; and, paradoxically, no practice is so universally neglected in modern Catholicism. From the tales of Abraham to the fasting of the Ninevites, who averted the wrath of God by their penance, to the tale of Sarah who fasted before her wedding to Tobias, to the words of our Lord that certain demons could only be overcome by fasting, Scripture is replete with examples of the importance of fasting and its efficacy in purifying the soul and obtaining God's favor; the lives of the saints afford us with thousands more examples. But how necessary is fasting to the advancement of the spiritual life, and what fruits do we derive from the practice? Let us examine these questions in light of Tradition, Scripture and the teaching of St. Robert Bellarmine.
Precept or Counsel?
It is first necessary to establish whether fasting is a precept or a counsel for Christians. Of course we are not speaking of those days of fast and abstinence that are mandated by Canon Law, but rather the practice of fasting as a form of spiritual asceticism. Is this sort of fasting a practice that is merely counseled, similar to celibacy, or is it a precept, such that it is such a necessity of the spiritual life that those who do not do it may gravely endanger the their soul?
It is an interesting fact of Christian history that the earliest of the Church Fathers are more interested in telling Christians when they should not fast than mandating when they should. For example, the Didache instructs Christians to avoid fasting on the same day as the Jews so as to avoid being lumped in with 'the hypocrites":
"Your fasts should not coincide with those of the hypocrites. They fast on Mondays and Tuesdays; you should fast on Wednesdays and Fridays." 
Thus, the earliest teaching on when Christians are to fast is determined by way of negation. But in ruling out certain days of the week, the Didache does in fact imply that Christians are expected to fast. This early tradition lived on in the practice of observing fasting during the Ember Days, which are the Wednesday, Friday, and Saturday after the First Sunday of Lent for spring, after Pentecost Sunday for summer, after the Feast of the Exaltation of the Cross (14th September) for autumn, and after the Third Sunday of Advent for winter. The implication in such days being set aside for fasting is that Christians will fast.
Our Lord Himself implied the same thing, and like the Didache, did so in the context of contrasting Christian fasting with the fasting of the Pharisees:
“Moreover, when you fast, do not be like the hypocrites, with a sad countenance. For they disfigure their faces that they may appear to men to be fasting. Assuredly, I say to you, they have their reward. But you, when you fast, anoint your head and wash your face, so that you do not appear to men to be fasting, but to your Father who is in the secret place; and your Father who sees in secret will reward you openly." 
Our Lord's attitude is one of presumption; He simply takes it for granted that Christians will fast, and not thinking it necessary to establish this point, He goes right on to discuss the manner in which Christians ought to fast.
Given the attitude of our Lord and the words of the Didache and other Christian writings on the subject, all of which presume fasting, and all of the examples of fasting afforded us by the Old and New Testaments and the lives of the saints, it seems safe to say that fasting is a necessary part of the spiritual life for any serious Christian; no Christian who avoids fasting is serious about his or her spiritual life.
Yet, it seems also that we must stop short of stating that fasting is a strict precept, in such a way that those who fail to fast beyond the prescribed times are guilty of a specific sin or will be damned for not fasting (and remember, we are speaking about fasting above and beyond the times of fasting prescribed by the Church, which most certainly are precepts). Thus, it can be said that fasting is a necessity, but only a general necessity, not a strict necessity. It is a necessity in the same way that reading the Scriptures is necessary; certainly, anyone who wants to get to heaven and advance in their spiritual life will read the Scriptures, for "ignorance of Scripture is ignorance of Christ", as the Catechism says - and, those who positively refuse to read the Scripture most likely endanger their soul. Even so, one could not say that reading the Bible is necessary for salvation in the strict sense, since people can go to heaven who have not or cannot read the Scriptures. Fasting seems to be in this general category; it is not strictly necessary, but you do jeopardize the resilience of your soul against sin and temptation if you positively refuse to fast. So, for all practical purposes, fasting is a necessity for any serious Christian.
Fruits of Fasting According to St. Robert Bellarmine
The reason fasting is so necessary because its fruits are so manifold, such that it becomes an indispensable aid on the journey to sanctity. St. John Chrysostom summarizes the importance of fasting quite succinctly when he says, "Fasting is the support of our soul: it gives us wings to ascend on high, and to enjoy the highest contemplation."  Let us look at the five fruits of fasting, according to St. Robert Bellarmine. 
1) Fasting Disposes the Soul for Prayer: In order to pray effectively, one must set the mind on things heavenly and pull our attentions and affections away from things merely earthly, which drag our minds and hearts down and serve as a barrier to contemplating divine things. Fasting aids us in detaching our attention from things temporal and disposes us to more effectually commune with God. Hence, Moses fasted for forty days in preparation for his communication with God on Mount Sinai ; Elijah similarly fasted for forty days and Daniel fasted for three weeks before receiving the series of visions that comprise the second half of the Book of Daniel.  Likewise our Lord fasted for forty days as He prepared Himself for his mission, and St. Francis of Assisi spent a month in prayer prior to receiving the holy stigmata. In the Church's liturgy, great feasts are traditionally preceded by periods of fasting, for it is evident from the examples of the Scriptures and the lives of the saints that we are better disposes to pray effectually when in a state of fasting.
Fasting Tames the Flesh: St. Paul admonishes us to "crucify the flesh, with its vices and concupiscences"  and offers himself as an example, saying, "I chastise my body, and bring it into subjection".  While the body is certainly not evil as the Manicheans taught, it nevertheless can become a distraction in the service of God because of the many bodily urges and passions that attempt to bend our will towards gratifying them. There are many bodily desires, but perhaps the most primal and fundamental bodily desire is the desire to eat. Hunger is experienced by all persons of both sexes, of all ages and all states in life. It is the fundamental bodily urge and is most indicative of our creaturely state and our contingent existence - it is the king of all bodily desires. Therefore, when we by fasting dethrone this king and force the urges of hunger to submit to the will, we progress greatly in taming the flesh and subjecting it to our reason. There is no means of subjecting the flesh that is more effectual than fasting.
Fasting Honors God: Besides this, fasting also gives honor and glory to God. This is because we ourselves, trained by asceticism, become living sacrifices that are pleasing to God.  This is why the Council of Nicaea, in Canon 5 calls the Lenten fast "a clean and solemn gift, offered by the Church to God." Pope St. Leo the Great also calls fasting a sacrifice: "For the sure reception of all its fruits, the sacrifice of abstinence is most worthily offered to God, the giver of them all."  Therefore fasting is a sacrifice that gives honor and glory to God.
Fasting is Penitential: Fasting is also a means of atoning for the punishment due to sin. This is related to what was said above regarding hunger as the king of the bodily desires. Because we are in the flesh, we all need food for nourishment, and thus the practice of fasting becomes inherently unpleasant; to effectively fast is to truly crucify the flesh, and though we can accustom ourselves to the practices, fasting itself is something that is difficult, unpleasant and requires virtue to do consistently. Thus, it becomes an act of penance, which we can offer to God in satisfaction for the penalty due to sins. The Scriptures and the Fathers give us many examples of this. The anger of God was averted by the fasting of the people of Nineveh, and the Jews in the days of Esther appeased God by prayer and fasting. Many citations from the Fathers could also be offered in support of this teaching, but let us offer only two: St. Cyprian of Carthage admonishes his people, "Let us appease the anger of an offended God by fasting and weeping, as He admonishes us."  St. Augustine also says, "No one fasts for human praise, but for the pardon of his sins" .
Fasting is Meritorious with God: Finally, fasting is meritorious with God, both in the sense that it is effective in obtaining favors from God, and in that fasting itself merits a divine reward. Hannah fasted and her prayers were heard by God because of her fasting, and she thus conceived the prophet Samuel; similarly, Sarah was delivered from a demon after fasting for three days. Our Lord warns us that certain demons can only be overcome with fasting.  If we return to the passage from the Gospel of Matthew cited above, we note that our Lord promises a reward for those who fast:
"But you, when you fast, anoint your head and wash your face, so that you do not appear to men to be fasting, but to your Father who is in the secret place; and your Father who sees in secret will reward you openly." 
So fasting assists us in obtaining God's assistance in our temporal affairs and also merits an eternal reward.
The Manner in Which We Ought to Fast
Now that we have examined the necessity of fasting and the fruits we derive from the practice, let us look at the manner in which we ought to fast, taking chiefly as our guides the words of the prophet Isaiah and the teaching of St. Robert.
In the book of the prophet Isaiah, the Israelites complain that, despite fasting in accordance with God's proscriptions, their prayers are not heard and they seemingly derive no benefit from their fasting. They say, ""Why do we fast, and you do not see it? Afflict ourselves, and you take no note of it?" 
In the following verses, God warns them that their fasts are unacceptable for a multitude of reasons. Let us look at His response:
"Lo, on your fast day you carry out your own pursuits, and drive all your laborers. Yes, your fast ends in quarreling and fighting, striking with wicked claw. Would that today you might fast so as to make your voice heard on high! Is this the manner of fasting I wish, of keeping a day of penance: That a man bow his head like a reed, and lie in sackcloth and ashes? Do you call this a fast, a day acceptable to the LORD? This, rather, is the fasting that I wish: releasing those bound unjustly, untying the thongs of the yoke; Setting free the oppressed, breaking every yoke; Sharing your bread with the hungry, sheltering the oppressed and the homeless; Clothing the naked when you see them, and not turning your back on your own." 
In the first place, though food is given up, God notes that they have not given up quarreling and fighting. The lesson is that the purpose of fasting is to mortify the flesh. The flesh cannot be mortified if it is denied in one manner but gratified in three others. The practice of fasting must go hand in hand with a general disposition of humility and penitence that accompanies all our actions, even those not strictly related to our fast. St. Robert Bellarmine says the same thing:
"Nor do they derive any fruit who, although they may eat more moderately, yet on fasting days, do not abstain from games, parties, quarrels, dissensions, lascivious songs, and immoderate laughter: and what is still worse, commit the same crimes as they would on ordinary days." 
God also chastises the Jews for the same reason our Lord does: they bend the head and lie in sackcloth and ashes so that it may be readily apparent to all that they are fasting. But what is true fasting? God says it ought to be an occasion for doing works of charity, clothing the naked, setting free the oppressed etc., and what's more, our Lord adds that this ought to be done with a clean appearance and fresh countenance, so it is not readily apparent that we are fasting.
The Church proposes fasting for us during the period of Lent, before certain great Feasts, during the Embertides, and during Advent. But this is all of a very nominal sort, two snacks and one meal not larger than the two snacks. Yet the example of the saints was much greater - many, like St. Francis, spent up to a third or a half of the year in periods of fasting. Even if our occupations or state in life do not make such radical practices possible, most of us could stand to fast a little more. After looking at all of the benefits we derive from it, what excuse could we possibly have not to?
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 Didache, 8
 Matt. 6:16-18
 St. John Chrysostom, Homily 1 on Genesis
 St. Robert Bellarmine, The Art of Dying Well (Sophia Institute Press: Manchester, NH, 2005), pp 57-64.
 Deut. 9:18
 1 Kings 19:18, Dan. 10:2-3
 Gal. 5:24
 1 Cor. 9:27
 Rom. 12:1
 St. Cyprian, On the Lapsed, 29
 St. Augustine, Sermon 60
 Mark 9:29
 Matt. 6:17-18
 Isa. 58:3
 Isa. 58:4-7
 Bellarmine, The Art of Dying Well, pg. 64