  Rectangular $N\times M$ matrix models can be solved in several qualitatively
distinct large $N$ limits, since two independent parameters govern the size of
the matrix. Regarded as models of random surfaces, these matrix models
interpolate between branched polymer behaviour and two-dimensional quantum
gravity. We solve such models in a `triple-scaling' regime in this paper, with
$N$ and $M$ becoming large independently. A correspondence between phase
transitions and singularities of mappings from ${\bf R}^2$ to ${\bf R}^2$ is
indicated. At different critical points, the scaling behavior is determined by:
i) two decoupled ordinary differential equations; ii) an ordinary differential
equation and a finite difference equation; or iii) two coupled partial
differential equations. The Painlev\'e II equation arises (in conjunction with
a difference equation) at a point associated with branched polymers. For
critical points described by partial differential equations, there are dual
weak-coupling/strong-coupling expansions. It is conjectured that the new
physics is related to microscopic topology fluctuations.
