This chapter examines random walks on the integer lattice in higher dimensions. A central result is Pólya’s theorem, which states that a symmetric random walk in the plane returns to its starting point with probability one, but this recurrence property is lost in three and higher dimensions. Another topic is the number of states visited by a random walk of a given length. It is shown that the relative proportion of the different visited states converges stochastically to the probability that the random walk does not return to the starting point. The chapter concludes with the discrete Dirichlet problem.