We study the statistics of the number of records R n for a symmetric, n-step, discrete jump process on a 1D lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution P (R n) of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival probability of a random walk on a line, with discrete or continuous jump distributions. For the discrete jump process, we then derive the asymptotic large n behavior of P (R n) as well as of the average number of records E(R n). We show that unlike the case of random walks with symmetric and continuous jump distributions where the record statistics is strongly universal (i.e., independent of the jump distribution for all n), the record statistics for lattice walks depends on the jump distribution for any fixed n. However, in the large n limit, we show that the distribution of the scaled record number R n /E(R n) approaches a universal, half-Gaussian form for any discrete jump process. The dependence on the jump distribution enters only through the scale factor E(R n), which we also compute in the large n limit for arbitrary jump distributions. We present explicit results for a few examples and provide numerical checks of our analytical predictions.