In this chapter, we prove weak convergence of locally perturbed random walks, properly scaled, to the processes discussed in the previous chapter. Our primary focus is on the one-dimensional case, and we consider two types of local perturbations. The first type is a random walk on nonnegative integers, reflected to the right upon crossing the origin. The second type is a random walk on integers, perturbed at a finite number of points. Under various assumptions regarding the distribution tails of both the increments of the original unperturbed random walk and the perturbations, we derive limit theorems for scaled versions of these processes in the Skorokhod space. The chapter closes with a discussion of some multidimensional perturbed random walks.