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I will discuss the long-time asymptotics (both of the moments and almost sure) of the solution to a parabolic second-order differential problem (so called parabolic Anderson model) on $\mathbb Z^d$ with a random i.i.d. potential, bounded from above. Both asymptotics are determined by appropriate variational principles. As an application, the Lifshitz tails for the spectrum of the associated random Schroedinger operator (Anderson Hamiltonian) can explicitly be computed. The results extend various findings about the "simple random walk among Poissonian obstacles" obtained by Sznitman and his school in the 1990s.