  We explicitly classify all pairs $(M,G)$, where $M$ is a connected complex
manifold of dimension $n\ge 2$ and $G$ is a connected Lie group acting properly
and effectively on $M$ by holomorphic transformations and having dimension
$d_G$ satisfying $n^2+2\le d_G<n^2+2n$. These results extend -- in the complex
case -- the classical description of manifolds admitting proper actions of
groups of sufficiently high dimensions. They also generalize some of the
author's earlier work on Kobayashi-hyperbolic manifolds with high-dimensional
holomorphic automorphism group.
