The main tool we use in this talk is the concept of joint spectral radius (j.s.r.) for a family of matrices. It turns out that the asymptotic behaviour of all the possible products of the matrices of a family (as the number of factors goes to infinity) is determined by the value of its j.s.r. In particular, the asymtotic stability is equivalent to the condition j.s.r.<1. It is known that the asymptotic behaviour of all the solutions of a linear difference equation may be described by studying the asymptotic behaviour of the products of the companion matrices associated to the difference equation. When the difference equation has constant coefficients, there is just one constant companion matrix, everything being easy and wellknown. In fact, it is sufficient to control the spectral radius of the companion matrix. On the contrary, when the coefficients are variable, the companion matrices may be even infinitely many and, in any case, they do not reduce to one constant matrix. Therefore, a satisfactory stability analysis may be done only by evaluating (or, at least, by approximating sufficiently well) the j.s.r. of the family formed by all of the companion matrices. Most of the progress about the theory of j.s.r. has been done in the last decade. This probably explains the fact that, at least at our knowledge, it has not yet been employed in the numerical applications. In this talk, after outlining some basic results concerning the j.s.r. available in the literature and some new results and conjectures of our own, we illustrate how the j.s.r. approach may be successfully applied to the zero-stability analysis of linear multistep methods for ordinary differential equations. As an example, we treat the case of the 3-step backward differentiation formula of order 3 with variable stepsize and obtain almost-optimal results. The j.s.r. approach has already been successfully applied in other disciplines (e.g., for the computation of the Holder exponent of certain wavelets). One of the main purposes of our talk is to promote this technique among numerical analysts and to show its potential in view of more general applications and harder stability analyses of numerical methods.