We say that a permutation $p$ contains the shorter permutation $q$ as a pattern
if $p$ contains $|q|$ entries, not necessarily in consecutive positions, 
whose pairwise relations to each other are the same as those of the entries of $q$. 
For instance, $p=3576241$ contains $q=231$, since the first, third and fifth entries of $p$
relate to each other as the entries of $q$, namely the leftmost entry is the second smallest, 
the middleone is the largest, and the rightmost entry is the smallest.

In the first part of this talk, we will review the early results of this 
fascinating and rapidly growing topic, including the celebrated Marcus-Tardos theorem 
from 2003. That theorem  shows that for any given pattern $q$, the number of
permutations of length $n$ that avoid $q$ is simply exponential, that is, there 
exists a constant $c_q$ so that $S_n(q)\leq c_q^n$. In the second part, we discuss 
some more recent developments, such recent results on the extremely tenacious pattern 1324, 
the disproof of old conjectures, and several open problems.