Some important Banach algebras are not Arens regular, i.e. the two canonical extensions of the multiplication given on the algebra to its bidual are different. Among the non-regular algebras (possessing a two- sided bounded approximate identity) there are some for which these so- called Arens products agree at least on certain (large) subspaces of the respective bidual. It turns out that the class of these Banach algebras (which we call Arens semi-regular) can be characterized resp. defined in terms of a certain embedding property of the space of double multipliers of the algebra with respect to its bidual. It is shown that all commutative and all regular Banach algebras are semi-regular; the algebra of compact operators on a Banach space X is semi-regular if the dual of X has the Radon-Nikodym property. \textit{V. Losert} and \textit{H. Rindler} have shown recently [Lect. Notes Math. 1064, 368-378 (1983)] that a group algebra \(L^ 1(G)\) is semiregular if and only if G is discrete or abelian. Therefore it seems reasonable to expect that also in other cases, semi-regularity of ''classical'' Banach algebras might correspond to certain ''classical'' properties of the objects being considered.