Matrices play an important role in both pure and applied mathematics. Matrices with entries in rings have been intensively studied and used; as well as matrices with entries in semirings, Boolean algebras, semigroups and lattices. This book focuses on a more general object: formal matrices or, alternatively, generalized matrices. What is a formal matrix? Without going into much detail, one might say that it is a matrix with elements in several different rings or bimodules.