Letv(t)t≥ 0, be the first time a perturbed random walk crosses a general non-linear boundary. We provide limit theorems for the first passage times, the stopped perturbed random walk and the overshoot ast→ ∞. In particular, we apply these results to the important case when the perturbed random walk is of the form, where is a random walk whose increments have positive, finite mean andgis positive, continuous and, possibly, has further smoothness properties. The traditional case considered in nonlinear renewal theory is when the summands have finite variance andgis twice continuously differentiable. We also prove some results concerning existence of moments and uniform integrability. A final section contains a number of examples.