We study the number of distinct sites visited by N random walkers after t steps SN(t) under the condition that all the walkers are initially at the origin. We derive asymptotic expressions for the mean number of distinct sites <SN(t)> in one, two, and three dimensions. We find that <SN(t)> passes through several growth regimes; at short times <SN(t)>~td (regime I), for tx<<t<<tx' we find that <SN(t)>~(t ln[N S1(t)/td/2])d/2 (regime II), and for t>>tx', <SN(t)>~NS1(t) (regime III). The crossover times are tx~ln N for all dimensions, and tx'~∞, exp N, and N2 for one, two, and three dimensions, respectively. We show that in regimes II and III <SN(t)> satisfies a scaling relation of the form <SN(t)>~td/2f(x), with x≡N<S1(t)>/td/2. We also obtain asymptotic results for the complete probability distribution of SN(t) for the one-dimensional case in the limit of large N and t.