The Y Combinator
The Y Combinator is a classic lambda calculus construct that many people find baffling. Here's my attempt to explain it as clearly as possible (no promises!). Familiarity with Haskell syntax is assumed.
The problem we're trying to solve is how to write an anonymous function (a "lambda") that is recursive. Normally, if you want to write a recursive function, it looks like this:
fac n = if n == 0 then 1 else n * fac (n-1)