The article discusses the gravitational collapse of massive objects and the resulting space-time singularities. It begins by noting that the discovery of quasistellar radio sources has renewed interest in gravitational collapse. Some authors suggest that the enormous energy emitted by these objects may result from the collapse of a mass of the order of $10^6$ to $10^8$ solar masses to within its Schwarzschild radius, accompanied by a violent release of energy, possibly in the form of gravitational radiation. However, the detailed mathematical discussion of such situations is difficult due to the complexity of general relativity. Most exact calculations have used the simplifying assumption of spherical symmetry, which precludes detailed discussion of gravitational radiation. The article then presents a detailed analysis of the gravitational collapse of a spherically symmetrical body. It notes that for a sufficiently great mass, there is no final equilibrium state. When sufficient thermal energy has been radiated away, the body contracts and continues to contract until a physical singularity is encountered at $r=0$. To an outside observer, the contraction to $r=2m$ appears to take an infinite time. Nevertheless, the existence of a singularity presents a serious problem for any complete discussion of the physics of the interior region. The article then discusses the question of whether this singularity is simply a property of the high symmetry assumed. It notes that the presence of perturbations which destroy the spherical symmetry could alter the situation. The recent rotating solution of Kerr also possesses a physical singularity, but since a high degree of symmetry is still present, it might again be argued that this is not representative of the general situation. Collapse without assumptions of symmetry will be discussed here. The article concludes with a detailed analysis of the existence of a trapped surface and the implications for the development of singularities in space-time. It notes that the existence of a singularity can never be inferred without an assumption such as completeness for the manifold under consideration. The article presents a series of assumptions and shows that they are together inconsistent. The article also discusses the implications of these findings for the understanding of gravitational collapse and space-time singularities.The article discusses the gravitational collapse of massive objects and the resulting space-time singularities. It begins by noting that the discovery of quasistellar radio sources has renewed interest in gravitational collapse. Some authors suggest that the enormous energy emitted by these objects may result from the collapse of a mass of the order of $10^6$ to $10^8$ solar masses to within its Schwarzschild radius, accompanied by a violent release of energy, possibly in the form of gravitational radiation. However, the detailed mathematical discussion of such situations is difficult due to the complexity of general relativity. Most exact calculations have used the simplifying assumption of spherical symmetry, which precludes detailed discussion of gravitational radiation. The article then presents a detailed analysis of the gravitational collapse of a spherically symmetrical body. It notes that for a sufficiently great mass, there is no final equilibrium state. When sufficient thermal energy has been radiated away, the body contracts and continues to contract until a physical singularity is encountered at $r=0$. To an outside observer, the contraction to $r=2m$ appears to take an infinite time. Nevertheless, the existence of a singularity presents a serious problem for any complete discussion of the physics of the interior region. The article then discusses the question of whether this singularity is simply a property of the high symmetry assumed. It notes that the presence of perturbations which destroy the spherical symmetry could alter the situation. The recent rotating solution of Kerr also possesses a physical singularity, but since a high degree of symmetry is still present, it might again be argued that this is not representative of the general situation. Collapse without assumptions of symmetry will be discussed here. The article concludes with a detailed analysis of the existence of a trapped surface and the implications for the development of singularities in space-time. It notes that the existence of a singularity can never be inferred without an assumption such as completeness for the manifold under consideration. The article presents a series of assumptions and shows that they are together inconsistent. The article also discusses the implications of these findings for the understanding of gravitational collapse and space-time singularities.