This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:(*)Pu+∂tu=f(x,t), for x∈Ω⊂Rn,t∈I⊂R. 1) For strongly elliptic pseudodifferential operators (ψdo's) P on Rn of order d∈R+, a symbol calculus on Rn+1 is introduced that allows showing optimal regularity results, globally over Rn+1 and locally over Ω×I:f∈Hp,loc(s,s/d)(Ω×I)⇒u∈Hp,loc(s+d,s/d+1)(Ω×I), for s∈R, 1<p<∞. The Hp(s,s/d) are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal (−Δ)a (0<a<1), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition suppu⊂Ω‾, the initial condition u|t=0=0, and f∈Lp(Ω×I), (*) has a unique solution u∈Lp(I;Hpa(2a)(Ω‾)) with ∂tu∈Lp(Ω×I). Here Hpa(2a)(Ω‾)=H˙p2a(Ω‾) if a<1/p, and is contained in H˙p2a−ε(Ω‾) if a=1/p, but contains nontrivial elements from daH‾pa(Ω) if a>1/p (where d(x)=dist(x,∂Ω)). The interior regularity of u is lifted when f is more smooth.