We investigate the stability properties of numerical methods for weakly singular Volterra integral equations of the second kind. Our theory extends the stability theory of linear multistep methods for ordinary differential equations. We introduce the concepts of A-stability, A(½π)-stability etc. for Abel-Volterra equations. The stability region is characterized in terms of the weights of the method. It is shown that the order of an A-stable convolution quadrature cannot exceed 2. Further we study the stability properties of implicit Adam methods, with particular emphasis on the question of A(½π)-stability.