1. In a set-associative cache memory system of the type wherein data lines are stored in a two-dimensional logical array of rows (called sets) and columns (called congruence classes) of storage cells and a main memory (real) address corresponds to each of the stored data lines and is stored in an associated directory, apparatus for predicting a set position in the logical array for access to a desired data line from a virtual address of the desired data line before determination of a real address corresponding thereto, comprising: a history table for storing set positions of past successful accesses to data lines in said set-associative cache, at least some of said stored set positions being stored each at an address in said history table determined by a virtual address of the data line corresponding thereto; and means for accessing said history table at an access address determined by a virtual address of a desired data line and retrieving a stored set position at said access address, said predicted set position being said retrieved set position.