People are rational, it is claimed, to the extent that they follow a set of antecedently-given normative principles for correct reasoning. In the realm of deductive reasoning these principles are (perhaps) what is given by classical deductive logic. In the realm of probabilistic reasoning they are (perhaps) what is mandated by some version of probability theory. Much of the recent writing on this topic stems from the work of Wason (in deductive reasoning) and Tversky & Kahneman (in probabilistic reasoning). (Although there are certainly earlier researchers with the same conclusions.) These researchers seem to have showed that people, even educated people; indeed, even people educated in the fields of deductive logic and statistics, are nowhere near the normative standards.
Much of the recent research has taken the direction of claiming either that people are not so bad (the experimental data has been misinterpreted, or the normative standard was incorrectly identified) or even that it is somehow conceptually impossible for people as a whole to be systematically irrational.
I will survey some of this work, focusing on various claims concerning "limited rationality", "bounded rationality", "efficiency of effort", "evolutionary psychology", and the like. In this realm it is common to cite facts to the effect that people will "do better" when information presented to them is in a certain format (e.g., Gigerenzer's advocacy of frequencies rather than probabilities) or that it is about a certain evolutionarily important feature (e.g., Cosmides & Tooby's attending to issues of "detecting cheaters" and the like). I will claim that none of these accounts really provide an answer to the initial challenges.
And then I will conclude with the claim that this whole literature has nothing whatsoever to do with the issue of irrationality anyway.