The present monograph is a state of the art survey of the geometry of matrices. Professor L. K. Hua initiated the work in this area in the middle forties. In this geometry, the points of the space are a certain kind of matrices of a given size, and the four kinds of matrices studied by Hua are rectangular matrices, symmetric matrices, skew-symmetric matrices and hermitian matrices. To each such space there is associated a group of motions, and the aim of the study is then to characterize the group of motions in the space by as few geometric invariants as possible. At first, Professor Hua, relating to his study of the theory of functions of several complex variables, began studying the geometry of matrices of various types over the complex field. Later, he extended his results to the case when the basic field is not necessarily commutative, discovered that the invariant "adjacency" alone is sufficient to characterize the group of motions of the space, and applied his results to some problems in algebra and geometry. Professor Hua's pioneer work in the area has been followed by many mathematicians, and more general results have been obtained. I think it is now time to summarize all results obtained so far, and this has been my motivation for the present work.