In the first part of this minicourse we give a survey on some aspects of general infinite-dimensional Lie theory. Here we put an emphasis on integrability problems (which are mostly trivial for finite-dimensional groups). A central question is how to decide for a locally convex Lie algebra if it integrates to an infinite-dimensional Lie group and to integrate morphisms of locally convex Lie algebras? Other related problems are: integrating Lie algebra extensions, integrating Lie subalgebras and integrating Lie algebras of vector fields.
In the second part we turn to some functional analytic issues of unitary representations of infinite-dimensional Lie groups. We discuss the class of semibounded representations (spectra of many infinitesimal generators are bounded below) and some of its basic properties. Finally we discuss a few classes of groups with a rich supply of semibounded representations.