We prove: (1) the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable, (2) the set of all Diophantine equations which have at most finitely many solutions in positive integers is not recursively enumerable, (3) the set of all Diophantine equations which have at most finitely many integer solutions is not recursively enumerable, (4) analogous theorems hold for Diophantine equations D(x₁, …, x_p) = 0, where p ∈ N\{0} and for every i ∈ {1, …, p} the polynomial D(x₁, …, x_p) involves a monomial M with a non-zero coefficient such that x_i divides M, (5) the set of all Diophantine equations which have at most k variables (where k ≥ 9) and at most finitely many solutions in non-negative integers is not recursively enumerable.