Chordal graphs play a fundamental role in algorithms for sparse matrix factorization, graphical models, and matrix completion problems. In matrix optimization chordal sparsity patterns can be exploited in fast algorithms for evaluating the logarithmic barrier function of the cone of positive definite matrices with a given sparsity pattern and of the corresponding dual cone. We will give a survey of chordal sparse matrix methods and discuss two applications in more detail: linear optimization with sparse matrix cone constraints, and the approximate solution of dense quadratic programs arising in support vector machine training.