In this book, we investigate the issue of the final fate of a gravitationally collapsing massive stars and the associated cosmic censorship problem. When the star is heavier than a few solar masses, it could undergo an endless gravitational collapse without achieving any equilibrium state. This happens when the star has exhausted its internal nuclear fuel which provides the outwards pressure against the inwards pulling gravitational forces. This remains one of the most outstanding open issues (Islam, 1992) in gravitation theory today. We discuss the essential features of spherically symmetric collapse and the final fate of a massive star which undergoes gravitational collapse. It is the spherically symmetric collapse of a homogenous dust cloud, as described by the Oppenheimer-Snyder model (1939), which led to the general concept of trapped surfaces collapsed to singularities. Our objective here is to discuss the above and such other related important questions concerning the global aspects of gravity, and also to discuss some applications of the results which are already available on the global structure of space-time singularity.