In this paper, we study the discrete-time nonlinear consensus protocols over both directed and undirected networks with fixed topology. First, the notions of (global/exponential) semistability are introduced for systems with a continuum of equilibria. In terms of (global/exponential) consensus defined based on the notion of semistability, we have derived convergence conditions for the general discrete-time distributed nonlinear consensus protocols over directed/undirected weighted networks. We also show that several convergence results on linear consensus problems in existing literature can be derived as special cases of our main result. In the end, a numerical example is given to illustrate our theoretical results.