We discuss the prospects for obtaining constraints on the equation of state from astrophysical sources. Neutron star masses although few are known at present, provide a very direct constraint in as much as the connection to the equation of state involves only the assumption that Einstein's general theory of relativity is correct at the macroscopic scale. If the millisecond pulses briefly observed in the remnant of SN1987 A can be attributed to uniform rotation of a. pulsar, then a. very severe constrajnt is placed on the equation of state. The theory again is very secure. The precise nature of the constraint is not yet understood, but it appears that the equation of state must be neither too soft nor stiff, and it may be that there is information not only on the stiffness of the equation of state but on its shape. Supernovae simulations involve such a plethora. of physical processes including those involved in the evolution of the precollapse configuration, not all of them known or understood, that they provide no constrajnt at the present time. Not even the broad category of mechanism for the explosion is agreed upon (prompt shock, delayed shock, or nuclear explosion). In connection with very fast pulsars, we include some speculations on pure quark matter stars, and on possible scenarios for understanding the disappearance of the fast pulsar in SN1987 A .