Consider a polynomial p(s, k) which is affine in the parameter k; assume that the vector k is uncertain but belongs to a bounded closed convex set which contains the origin as an interior point, and call a polynomial stable if all of its roots are contained in a pre-specified stability region in the complex plane. Then the stability robustness of p(s, k) can be measured by the maximal nonnegative number with the property that if the gauge (or the Minkowski functional) of k with respect to the convex set is less than , the polynomial p(s, k) is always stable. This paper develops a unified approach to compute the robustness measure using the framework of convex analysis. The approach is very general because of two reasons: (i) the stability region in the complex plane can be an arbitrary open set, which includes the open left half plane and the open unit disc as special cases; (ii) the convex set in which k is contained can be assumed to have an arbitrary shape, which includes polytopes and ellipsoids as special cases. The computational procedure to compute which results from this approach is easy to implement. An example is included to illustrate the type of results which may be obtained. The approach developed in this paper, when specialized, leads to some of the available results in the literature; moreover, it generates many new and interesting results. Another important feature of this approach is that it provides a rich mathematical insight to the stability robustness problem of polynomials.