Abstract: The problem of understanding the distribution of the length of the longest increasing subsequence in a random permutation has a surprisingly rich mathematical structure. It turns out that the asymptotic fluctuations are the same as for the largest eigenvalue of a large hermitian random matrix. The methods involved in understanding this problem comes from combinatorics (RSK-correspondence, symmetric polynomials), random matrix theory and asymptotic analysis. The problem and its generalizations have connections with interesting probability problems concerning for example random growth and first/last-passage percolation.