Abstract <p>The first boundary value problem for the heat equation in a cone is considered in the case of a domain degenerating at the initial time. The eigenfunctions of the problem are found. Estimates of the Green’s function are obtained. For the problem with a zero boundary function, unique solvability in some class of functions admitting certain growth as they approach the vertex of the cone is established. Similar results are obtained for a cone degenerating at the final moment of time. Additionally, the first boundary value problem in domains degenerating only with respect to some of the variables is con-sidered.</p>