In 1955 Erdős posed the multiplication table problem: Given a large integer N, how many distinct products of the form ab with a ≤ N and b ≤ N are there? The order of magnitude of the above quantity was determined by Ford. The purpose of this thesis is to study generalizations of Erdős's question in two different directions. The first one concerns the k-dimensional version of the multiplication table problem: for a fixed integer k ≥ 3 and a large parameter N, we establish the order of magnitude of the number of distinct products n₁···n _k with n₁ ≤ N for all i ∈ {1,...,k}. The second question we shall discuss is the restricted multiplication table problem. More precisely, for [special characters omitted] we seek estimates on the number of distinct products ab ∈ [special characters omitted] with a ≤ N and b ≤ N. This problem is intimately connected with the available information on the distribution of [special characters omitted] in arithmetic progressions. We focus on the special and important case when [special characters omitted] = P_s = {p + s : p prime} for some fixed s ∈ [special characters omitted] \ {0}. Ford established upper bounds of the expected order of magnitude for |{ab ∈ P_s : a ≤ N, b ≤ N}|. We prove the corresponding lower bounds thus determining the size of the quantity in question up to multiplicative constants.