A new, general relativistic, hydrodynamical computer code for spherical, stellar colapse to neutron stars and black holes is presented. The code is employed to construct spacetimes for the geometrodynamical evolution of spherical masses which are initially at rest and obey an adiabatic GAMMA-law equation of state. Collapse to a black hole can be followed accurately; the evolution outside the event horizon can be determined far into the future without having the integration terminate because of spacetime singularities inside.We use the code to address several fundamental points of principle regarding spherical, gravitational collapse and requiring a reliable, fully relativistic, numerical computation. We sample a variety of possible collapse and requiring a reliable, fully relativistic, numerical computation. We sample a variety of possible collapse scenarios, characterized by different masses, mass distributions, and equations of state, to examine stellar core collapse and its end fate. We distinguish three possible outcomes for adiabatic core collapse: homologous core, bounce, nonhomologous core bounce, and catastrophic collapse to a black hole. Even a very massive stellar core may undergo a nonhomologous bounce and only later collapse to a black hole on a secular time scale rather than a dynamical one. Accretion-driven neutron star collapse is also treatedmore » numerically. The implications of our findings for realistic core collapse believed to occur in nature are considered.« less