Finite-time Cosmological Singularities and the Possible Fate of the Universe

Singularities in any physical theory are either remarkable indicators of the unknown underlying fundamental theory, or indicate a change in the description of the physical reality. In General Relativity there are three fundamental kinds of singularities that might occur, firstly the black hole spacelike crushing singularities, e.g. in the Schwarzschild case and two cosmological spacelike singularities appearing in finite-time, namely, the Big Bang singularity and the Big Rip singularity. In the case of black hole and Big Bang singularity, the singularity indicates that the physics is no longer described by the classical gravity theory but some quantum version of gravity is probably needed. The Big Rip is a future singularity which appears in the context of General Relativity due to a phantom scalar field needed to describe the dark energy era. Apart from the Big Rip singularity, a variety of finite-time future singularities, such as, sudden singularity, Big Freeze singularity, generalized sudden singularity, $w$-singularity and so on, are allowed in various class of cosmological models irrespective of their origin. The occurrence of these finite-time singularities has been intensively investigated in the context of a variety of dark energy, modified gravity, and other alternative cosmological theories. These singularities suggest that the current cosmological scenario is probably an approximate version of a fundamental theory yet to be discovered. In this review we provide a concrete overview of the cosmological theories constructed in the context of Einstein's General Relativity and modified gravity theories that may lead to finite-time cosmological singularities. We also discuss various approaches suggested in the literature that could potentially prevent or mitigate finite-time singularities within the cosmological scenarios.