The book gives a description of the principal models of quantum gravity which are based on a state sum for a piecewise linear (PL) spacetime manifold corresponding to a triangulation of a smooth spacetime manifold. There are detailed descriptions of Regge quantum gravity, spin-foam and spin-cube models, while the other models, like causal dynamical triangulations, are only briefly described, since the sum over triangulations is outside the scope of the book.More details are given for the approach where the PL manifold is considered as a basic structure of spacetime, instead of being used as an auxiliary tool to define the gravitational path integral for a smooth manifold. One of the advantages of a PL spacetime is that it is straightforward to construct a finite gravitational path integral. Another advantage is that the problem of determination of the classical limit and calculation of the quantum corrections is solved by using the effective action method. Also, the smooth manifold limit becomes a smooth manifold approximation, since a smooth manifold can be approximated by a PL manifold which contains many simplicial cells.Some physical effects of a QG theory on a PL spacetime are described, one of which is that the cosmological constant spectrum contains the observed value.The book also contains a short exposition of higher gauge theory, which is an interesting way to generalize a gauge symmetry by using the concept of a 2-group, which is a categorical generalization of a group. This approach gives the spin-cube models.Key FeaturesDescribes a novel approach in the study of quantum gravity (QG) state-sum models, which is based on the application of the effective action method from quantum field theory. Related to that is a study of the effect of a non-trivial path-integral (PI) measure on the PI finiteness, as well as a study on the dependence of the semi-classical expansion of the effective action on the PI measureDetailed study of the idea that the spacetime at small distances is not a smooth manifold but a piecewise linear (PL) manifold corresponding to a triangulation of a smooth manifold. One consequence of a QG theory on a PL spacetime is that the cosmological constant has a continuous spectrum, and the spectrum contains the observed value of the cosmological constantA clear explanation of the relationship between a higher gauge theory and general relativity, and how new state-sum models of QG are then constructed. These new state-sum models are called spin-cube models, which are categorical generalizations of the spin-foam models, since one labels the edges, triangles and tetrahedra with representations of 2-groups, which are categorical generalizations of groups

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