The mathematical background to the topic of asymptotic approximation methods for the solution of ordinary and partial differential equations is given first. These methods are then applied to several examples of problems from astrophysical fluid (and magneto-fluid) dynamics. An entire chapter is devoted to each topic and among them are - accretion disk boundary layers, fronts in thermally bistable interstellar medium; ideas from geophysics (shallow water theory) applied to modeling accretion disks; the discovery of transient perturbation growth in accretion disks, a growth than may reasonably give rise to secondary instabilities which, in turn, can be instrumental in driving angular momentum transport in these objects. Finally a critical nonlinear analysis of the magnetic Taylor-Couette-flow is given. It is found that for the latter, the magneto-rotational instability, whose importance for driving accretion disk turbulence had been considered paramount, actually saturates at an amplitude that goes with the magnetic Prandtl number, a fact that would imply that this instability, per- se, cannot cause sufficient angular momentum transport to induce accretion.