This talk will be an introduction to permutation patterns. A permutation p in Sn is said to contain a pattern permutation q in Sm if there exists a subsequence of p that is order-isomorphic to q. If p does not contain q, then p avoids q. One of the central questions in the study of patterns is: given a pattern p in Sm, how many permutations in Sn avoid p? This question has a beautiful answer for the case m=3 and we will prove the result in this case. When m=4, the problem gets extraordinarily more complicated. We will survey what is known for m=4 and discuss what is still left to be solved.
