  We describe a powerful methodology for numerical solution of 3-D
self-gravitational hydrodynamics problems with extremely high resolution. Our
method utilizes the technique of local adaptive mesh refinement (AMR),
employing multiple grids at multiple levels of resolution. These grids are
automatically and dynamically added and removed as necessary to maintain
adequate resolution. This technology allows for the solution of problems in a
manner that is both more efficient and more versatile than other fixed and
variable resolution methods. The application of AMR to simulate the collapse
and fragmentation of a molecular cloud, a key step in star formation, is
discussed. Such simulations involve many orders of magnitude of variation in
length scale as fragments form. In this paper we briefly describe the
methodology and present an illustrative application for nonisothermal cloud
collapse. We describe the numerical Jeans condition, a criterion for stability
of self-gravitational hydrodynamics problems. We show the first well-resolved
nonisothermal evolutionary sequence beginning with a perturbed dense molecular
cloud core that leads to the formation of a binary system consisting of
protostellar cores surrounded by distinct protostellar disks. The scale of the
disks, of order 100 AU, is consistent with observations of gaseous disks
surrounding single T-Tauri stars and debris disks surrounding systems such as
$\beta$ Pictoris.
