The main result of this dissertation concerns the asymptotics, uniform in t and x, of the probability distribution of a random walk with heavy tails. The random walk is a Markov process and thus can be characterized in terms of their generators. We impose certain conditions on the Fourier transform of the kernel of the generator, which still allow us to consider rather general class of processes on Zd. The process we consider can be viewed as a generalization of the simple symmetric walk (in continuous time) for which both the central limit theorem and large deviation results are well-known. For problems with heavy tails, the analogue of the central limit theorem is the convergence of the properly normalized process to the stable laws. In terms of probability densities, these limit theorems give the asymptotics of p(t,x,0) when x is of order t1/á. For the class of random walk under consideration, we obtain the asymptotics of p(t,x,0) uniformly in t and x for all t>1, x ϵ Rd, covering, in particular, the regime of the central limit theorem and large deviations.