The discovery of quasistellar radio sources has reignited interest in gravitational collapse. Some suggest that the immense energy emitted by these objects may result from the collapse of a mass of 10^6 to 10^8 solar masses to within its Schwarzschild radius, releasing energy possibly as gravitational radiation. However, general relativity's complexity makes detailed analysis difficult, so most studies assume spherical symmetry, which limits discussion of gravitational radiation. For a sufficiently massive, spherically symmetric body, there is no final equilibrium state. As thermal energy is radiated away, the body contracts until it reaches a physical singularity at r=0. To an outside observer, this contraction appears to take infinite time. The existence of a singularity poses a serious problem for understanding the interior physics. The question arises whether this singularity is merely a result of high symmetry. While radial collapse to a single point may seem unsurprising, perturbations could alter this. The Kerr solution, though symmetric, still has a physical singularity, but it may not represent general cases. This paper discusses gravitational collapse without assuming symmetry. It considers the time development of a Cauchy hypersurface representing an initial matter distribution. Assuming Einstein's equations and suitable equations of state, it shows that deviations from spherical symmetry cannot prevent spacetime singularities. If actual physical singularities are not allowed, then one of the following must hold: negative local energy, violation of Einstein's equations, incomplete spacetime manifold, or loss of spacetime meaning at high curvatures. A spherically symmetric matter distribution collapsing symmetrically leads to a Schwarzschild field. A trapped surface, defined as a closed, spacelike surface with converging null geodesics, exists if the matter region has no sharp boundary or if spherical symmetry is dropped, provided deviations are not too great. The existence of a trapped surface implies singularities necessarily develop. However, without assuming manifold completeness, singularities cannot be inferred. The paper assumes the manifold is null complete into the future. It shows that these assumptions are inconsistent. The existence of a trapped surface leads to a caustic, making the boundary compact. Approximating this boundary with a smooth, closed, spacelike hypersurface, it shows that the degree of the map must be unity, which is impossible due to the noncompactness of the Cauchy hypersurface. Full details of these results will be published elsewhere.The discovery of quasistellar radio sources has reignited interest in gravitational collapse. Some suggest that the immense energy emitted by these objects may result from the collapse of a mass of 10^6 to 10^8 solar masses to within its Schwarzschild radius, releasing energy possibly as gravitational radiation. However, general relativity's complexity makes detailed analysis difficult, so most studies assume spherical symmetry, which limits discussion of gravitational radiation. For a sufficiently massive, spherically symmetric body, there is no final equilibrium state. As thermal energy is radiated away, the body contracts until it reaches a physical singularity at r=0. To an outside observer, this contraction appears to take infinite time. The existence of a singularity poses a serious problem for understanding the interior physics. The question arises whether this singularity is merely a result of high symmetry. While radial collapse to a single point may seem unsurprising, perturbations could alter this. The Kerr solution, though symmetric, still has a physical singularity, but it may not represent general cases. This paper discusses gravitational collapse without assuming symmetry. It considers the time development of a Cauchy hypersurface representing an initial matter distribution. Assuming Einstein's equations and suitable equations of state, it shows that deviations from spherical symmetry cannot prevent spacetime singularities. If actual physical singularities are not allowed, then one of the following must hold: negative local energy, violation of Einstein's equations, incomplete spacetime manifold, or loss of spacetime meaning at high curvatures. A spherically symmetric matter distribution collapsing symmetrically leads to a Schwarzschild field. A trapped surface, defined as a closed, spacelike surface with converging null geodesics, exists if the matter region has no sharp boundary or if spherical symmetry is dropped, provided deviations are not too great. The existence of a trapped surface implies singularities necessarily develop. However, without assuming manifold completeness, singularities cannot be inferred. The paper assumes the manifold is null complete into the future. It shows that these assumptions are inconsistent. The existence of a trapped surface leads to a caustic, making the boundary compact. Approximating this boundary with a smooth, closed, spacelike hypersurface, it shows that the degree of the map must be unity, which is impossible due to the noncompactness of the Cauchy hypersurface. Full details of these results will be published elsewhere.