Existence of classical solution for Poisson's equation

Question

I read a bit through uniqueness of the solution to Poisson's equation with boundary data and also existence/uniqueness of the variational formulation. But under what conditions can one guarantee the existence of a classical solution for Poissons equation $$-\Delta u =f$$ with given boundary data, say dirichlet, $\Omega \subset \mathbb R^n$?