  The occurrence of a spacetime singularity indicates the breakdown of Einstein
gravitation theory in these extreme regimes. We consider here the singularity
issue and various black hole paradoxes at classical and quantum levels. It is
pointed out that a possible resolution to these problems could be arrived at by
avoiding the formation of trapped surfaces during a continual gravitational
collapse. A class of perfect fluid collapse models is constructed which
realizes such a possibility. While the pressure could be negative in the
interior of the cloud, the weak energy condition is satisfied. The collapsing
star radiates away most of its matter as the process of gravitational collapse
evolves, so as to avoid the formation of trapped surfaces and the spacetime
singularity. The collapsing interior is matched to an exterior which is a
generalized Vaidya spacetime to complete the model.
