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.