Reason is almost as vague a term as its Greek cognate "logos." Famously mistranslated as "word," the Greek original and the modern English term agree that a reason is an account or story about the causes behind some phenomenon or event.
There have been many attacks on Reason, especially since the failures of "modern" theology in the Middle Ages (by Islam, Judaism, and the Scholastics in that order) to reason to God, and since the Enlightenment failure to reason to human nature and morality.
On one level, a reasoned argument against Reason would appear to be the same vicious circle that led the ancient skeptics to avoid claiming that all knowledge was relative. On another level, the best arguments will rediscover the virtuous circle, the power of Reason inside Ideal Systems, and the natural limits of Reason alone to account for things in the world.
Since the separation of Science (Natural Philosophy) from Philosophy proper, Reason has come to describe the Scientific Method, with its additions of hypothesis and experimental test to the original sense of Reason as logical and deductive thought. Some thinkers erroneously included inductive thought.
The Problem of Induction arises when repeated patterns of events suggest an explanation in terms of causality. If A has always been followed by B, then it suggests that A causes B. This fails because inductive logic alone can never produce knowledge about the contingent world. The correct view is that an inductive pattern may lead to a hypothesis (a theory), which can be experimentally tested. Note that mathematical induction (really a form of deduction?) is a valid form of reasoning.
The identification of Reason with Science led Immanuel Kant to criticize Reason. The fullness of Newton's theories to account for the motions of the entire universe, especially their implied determinism, were unacceptable to Kant, who wanted to make room for human freedom, values, God, and immortality. Romantics of the nineteenth and twentieth centuries went "beyond Reason" to intuition, hermeneutics, to art and the irrational in search of meaning deeper than science alone can provide.