The construction of free Banach-Lie algebra over a normed space enables us to build a connected separable Banach-Lie group of which any other connected separable Banach-Lie group is a quotient. New proofs are given to the result on representabil ity of any Banach-Lie algebra as a quotient of an enlargable Banach-Lie algebra (due to van Est and Swierczkowski) and to the result on representabil ity of any topological group as a quotient of a group with no small subgroups (due to successive efforts of Morris and Thompson, the author, and Sipacheva and Uspenskiϊ). 1. Introduction. Over the last 50 years a number of constructions of universal arrows (see, e.g., [Go]) to the categories of topological algebraic systems have been studied. Important contributions are those by Markov [M], Graev [Gr], and Arhangel'skiϊ [A2] on free topological groups, Mal'cev [Me] on free topological algebras, Arens and Eells [AE], Raϊkov [R], and Uspenskiϊ [U] on free Banach spaces and free locally convex spaces. By virtue of these constructions a first ever example of a non-normal Hausdorff topological group was obtained [M], and the representability of any topological group as a quotient group of a zero-dimensional group was proved [Al]. Here we apply the concept of a free complete normed Lie algebra to theory of topological and Lie groups. Our construction is an extension of the well-known construction of Arens-Eells [AE] to the case of normed Lie algebras. Our main result is that there exists a couniversal separable connected Banach-Lie group, that is, such a separable connected Banach-Lie group that any other such Banach-Lie group is its quotient Lie group. This follows from observation that any free Banach-Lie algebra is enlargable, that is, comes from an approriate Banach-Lie group. Also we give entirely new and rather transparent proofs of two earlier known results. Cohomological technique has enabled van Est and independently Swierczkowski [S2] to prove that any Banach-Lie algebra is a quotient algebra of an enlargable Banach-Lie algebra. Here we deduce the result from enlargability of free Banach-Lie algebras. In his book [Ka] Kaplansky asked whether a quotient group of a