Not much is known about theweight distribution of the generalized Reed-Muller code RMq (s,m) when q > 2, s > 2 and m ≥ 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m - 1)(q - 1) we then find the first s +1-(m -1)(q -1) weights. For the case m = 2 the second weight is now known for all +1--1-1weights. For the case = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.