  A topological group is called a pro-Lie group if it is isomorphic to a closed
subgroup of a product of finite-dimensional real Lie groups. This class of
groups is closed under the formation of arbitrary products and closed subgroups
and forms a complete category. It includes each finite-dimensional Lie group,
each locally compact group which has a compact quotient group modulo its
identity component and thus, in particular, each compact and each connected
locally compact group; it also includes all locally compact abelian groups.
This paper provides an overview of the structure theory and Lie theory of
pro-Lie groups including results more recent than those in the authors'
reference book on pro-Lie groups. Significantly, it also includes a review of
the recent insight that weakly complete unital algebras provide a natural
habitat for both pro-Lie algebras and pro-Lie groups, indeed for the
exponential function which links the two. (A topological vector space is weakly
complete if it is isomorphic to a power $\R^X$ of an arbitrary set of copies of
$\R$. This class of real vector spaces is at the basis of the Lie theory of
pro-Lie groups.) The article also lists 12 open questions connected with
pro-Lie groups.
