An increasing subsequence of a permutation a1, a2, …, an of 1, 2, …, n is a subsequence ai(1) < ai(2) < … < ai(k) (so 1 ≤ i(1) < i(2) < … < i(k) ≤ n); a decreasing subsequence is defined similarly. We will survey the theory of increasing and decreasing subsequences. Topics will include the connection with Young tableaux and the Robinson-Schensted-Knuth (RSK) algorithm, the expected length of the longest increasing subsequence of a random permutation of 1, 2, …, n (due to Logan-Shepp and Vershik-Kerov), the limiting distribution of the length of the longest increasing subsequence (due to Baik-Deift-Johansson), and some variations concerned with alternating subsequences, matchings, etc.