We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in non-critical harmonic lattice system of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape and states of nonzero temperature. For nearest-neighbor interactions - corresponding to the Klein-Gordon case - upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of [Phys. Rev. Lett. 94, 060503 (2005)]. These higher dimensional analogues of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in non-critical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement.