In this talk we will survey some open problems and recent results in probabilistic and extremal combinatorics related to finding perfect matchings in hypergraphs. The general plan (if time permits) is to motivate the general problem, discuss it for graphs, explain the difficulty when passing to hypergraphs, relate the basic problem to an unsolved probabilistic inequality which was conjectured by Feige, sketch some useful tools for tackling these problems, and report on some recent developments in the area.