We define a class of ${\mathbb{Z}}^d$-actions, d ≥ 2, called product ${\mathbb{Z}}^d$-actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing ${\mathbb{Z}}^d$-action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic ${\mathbb{Z}}^d$-action with Lebesgue component of multiplicity $2^{d}k$, where k is an arbitrary positive...