In this memoir, we shall consider weighted convolution algebras on discrete groups and semigroups, concentrating on the group (ℚ,+) of rational numbers, the semigroup $(ℚ^{+•},+)$ of strictly positive rational numbers, and analogous semigroups in the real line ℝ. In particular, we shall discuss when these algebras are Arens regular, when they are strongly Arens irregular, and when they are neither, giving a variety of examples. We introduce the notion of 'weakly diagonally bounded' weights, weakening the known concept of 'diagonally bounded' weights, and thus obtaining more examples. We shall also construct an example of a weighted convolution algebra on ℕ that is neither Arens regular nor strongly Arens irregular, and an example of a weight ω on $ℚ^{+•}$ such that $lim inf_{s→ 0+}ω(s) =0$.