Let \(\sim\) be an equivalence relation on a matrix space M. Using a uniform approach, linear operators T: \(M\to M\) that preserve \(\sim\), that is, T(A)\(\sim T(B)\) whenever \(A\sim B\), are characterized for several important equivalence relations. These relations include consimilarity on the set of all \(n\times n\) complex matrices (A\(\sim B\) means \(A=\bar SBS^{-1}\) for some nonsingular matrix S), *-congruence on the set of all \(n\times n\) Hermitian matrices or on the set of all \(n\times n\) complex matrices, and unitary equivalence on the set of all \(n\times n\) complex or real matrices. One result: Let M be the set of all \(n\times n\) complex matrices. A linear operator T: \(M\to M\) preserves consimilarity if and only if there exists a nonsingular S and a real number \(\alpha\geq 0\) such that either \(T(X)=\alpha \bar SX^ tS^{-1}\) for all \(X\in M\) or \(T(X)=\alpha \bar SXS^{-1}\) for all \(X\in M\). (Here \(X^ t\) denotes the transpose of X.)