A matrix nearness problem consists of finding, for an arbitrary matrix A, a

nearest member of some given class of matrices, where distance is measured in a

matrix norm. A survey of nearness problems is given, with particular emphasis

on the fundamental properties of symmetry, positive definiteness, orthogonality,

normality, rank-deficiency and instability. Theoretical results and computational

methods are described. Applications of nearness problems in areas including control

theory, numerical analysis and statistics are outlined.