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We establish sharp interior and boundary regularity estimates for solutions to ∂ t u − L u = f ( t , x ) in I × Ω , with I ⊂ R and Ω ⊂ R n . The operators L we consider are infinitesimal generators of stable Lévy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that u is C 2 s + α in x and C 1 + α 2 s in t , whenever f is C α in x and C α 2 s in t . In the case f ∈ L ∞ , we prove that u is C 2 s − ϵ in x and C 1 − ϵ in t , for any ϵ > 0 . On the other hand, we study the boundary regularity of solutions in C 1 , 1 domains. We prove that for solutions u to the Dirichlet problem the quotient u / d s is Hölder continuous in space and time up to the boundary ∂Ω, where d is the distance to ∂Ω. This is new even when L is the fractional Laplacian. [ABSTRACT FROM AUTHOR]