I discuss in talk the Cauchy problem for fully nonlinear and possibly degenerate parabolic equations of the form ut + F(t, x, u,Du,D2u) = 0 set in QT = (0, T] × Rn. The delicate uniqueness issue is the main topic. Since the PDE is set in an unbounded set, it is, generally speaking, necessary to impose restrictons on the growth on the solution to prove uniqueness. However, I mention uniqueness results without any restrictions, for (i) the inviscid Hamilon-Jacobi equation ut + H(t, x,Du) = 0 or for (ii) the viscous equation ut+ 1 2 |Du|2+V (x)-"u = 0. A result for the Isaacs equation under an exponential growth condition on the solution is also given.