In this talk we address the problem of the computation of the joint spectral
radius - in short the j.s.r. - of a set of matrices, which is a recent research
topic.

First we briefly describe the extension of the spectral radius from a single
matrix to a set of matrices and illustrate some applications where such concept
plays an important role.

Then we pass to consider the problem of the computation of the j.s.r.  and
illustrate some possible strategies, mainly focusing on one we have recently
developed.  A basic tool we use to this purpose consists of polytope norms,
both real and complex. A complex polytope norm is a norm whose unit ball is
given by a balanced complex polytope, that is the extension of a classical
centrally symmetric real polytope to the complex space.  In order to describe
this kind of norms we summarize the main geometric properties of a balanced
complex polytope.

Finally we illustrate a possible algorithm for the computation of the j.s.r. of
a family of matrices which is based on the use of these classes of norms.  Some
examples will be shown to illustrate the behaviour of the algorithm.

Before concluding we address the problem of finite computability of the j.s.r.
and state some recent results, open problems and conjectures connected with
this issue.