Computing the Frobenius Canonical Form Arne Storjohann University of Western Ontario, London, Ontario, Canada Computing the Frobenius canonical form of a matrix over a field is a classical problem with many applications. The form is entirely rational and thus well suited to be computed symbolically; this is in contrast to the closely related Jordan form which may have entries --- eigenvalues of the input matrix --- in an algebraic extension of the ground field. In the first part of the talk I will define the form and recall some of it's properties and uses. The complexity of computing the form has been well studied, with first efficient algorithm given about fifteen years ago by Ozello. Many algorithms have been proposed since then; in the second part of the talk I will give an overview. Special attention will be given to recently discovered deterministic algorithms which recover the form in about the same number of field operations as required matrix multiplication --- this is nearly optimal. I will conclude by mentioning some open problems and recent results with respect to the problems of computing the form for a sparse input matrix and/or integer input matrix.