We discuss a variety of matrix distance problems. For a given square complex matrix A these include the nearest singular matrix, the nearest unstable matrix, and the nearest matrix with multiple eigenvalues. For a given pair A and B, we also consider (for A square) the nearest uncontrollable pair as well as (for both A and B square) the nearest pair with a common eigenvalue. We restrict our attention to the spectral and Frobenius norms, and emphasize the crucial role of pseudospectra. Finally, we discuss some structured nearest matrix problems. The issues include both characterizations of solutions and algorithms for finding them.