We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance σ^2 of the jump distribution is finite and (II) when σ^2 is divergent as in the case of Lévy flights with index 0 < μ< 2. In both cases we find that the mean record number <R_{n,N}> grows universally as \sim α_N \sqrt{n} for large n, but with a very different behavior of the amplitude α_N for N > 1 in the two cases. We find that for large N, α_N \approx 2 \sqrt{\log N} independently of σ^2 in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, α_N \approx 4/\sqrtπ, independently of 0<μ<2. For finite σ^2 we argue, and this is confirmed by our numerical simulations, that the full distribution of (R_{n,N}/\sqrt{n} - 2 \sqrt{\log N}) \sqrt{\log N} converges to a Gumbel law as n \to \infty and N \to \infty. In case II, our numerical simulations indicate that the distribution of R_{n,N}/\sqrt{n} converges, for n \to \infty and N \to \infty, to a universal nontrivial distribution, independently of μ. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poors 500 index.