The authors deal with techniques for the approximate solution of Volterra equations of the first kind of Hammerstein type with convolution kernels, \(\int_0^t k(t-s) S(s,u(s)) ds = f(t)\). This includes the case of linear equations as a special case, and this special case is discussed in great detail. The integral operator is assumed to satisfy a smoothing condition, i.e.the kernel \(k\) has a zero of a certain order at the origin. This leads to a certain amount of ill-posedness of the problem, and this is tackled by a local regularization approach (to be precise: sequential Tikhonov regularization and future polynomial regularization). The advantage of such an approach is that causality is retained. Specifically the authors answer questions for stability and convergence of the methods in an affirmative way. A number of illustrative examples is given.