In this paper, we solve the O-Neumann problem on (0,q) forms, q > 0, in the strictly pseudoconvex Siegel domain. The metric we use is invariant under the action of the Heisenberg group on the domain. We give a formula for a kernel of the Neumann operator, the fundamental solution of the problem, in (1.6) and Theorem 4.2, and we prove uniqueness of the Neumann operator. As far as we know, this is the first example of an explicit solution of the ~-Neumann problem. Greiner and Stein [GS] constructed a parametrix for the problem on (0, 1) forms in bounded strictly pseudoconvex domains. Phong [P] has announced an explicit construction of a parametric on (03) forms in the Siegel domain in C "+ 1, n > 1. Harvey and Polking [HP] have since constructed a Neumann kernel for the ~-Neumann problem on (p, q) forms on the ball. We obtain the kernel of the Neumann operator by integrating the fundamental solution of the corresponding heat equation ([S1], [$2]) with respect to time. To carry out the integration, we require estimates on the fundamental solution for large time. We did not need such estimates in [$1] and [$2]. Also, to prove that the resulting kernel is locally integrable, we require more precise estimates for small time. In Sect. 1, after some preliminaries, we state our main theorem. We then summarize the results of [S1 ] and [$2]. We solve the corresponding elliptic problem on C" + 1 in Sect. 2. In Sect. 3 we study the operator which corrects the boundary values of the solution we obtained in Sect. 2. We solve the O-Neumann problem in Sect. 4. Most of the work for this paper was done while I was a member of the Institute for Advanced Study. I wish to thank the Institute for its hospitality.