Introduction

The field of non-perturbative and background-independent quantum gravity has progressed considerably over the past few decades [78]. New research directions are being developed, new important developments are taking place in existing approaches, and some of these approaches are converging to one another. As a result, ideas and tools from one become relevant to another, and trigger further progress. The group field theory (GFT) formalism [39, 77, 79] nicely captures this convergence of approaches and ideas. It is a generalization of the much studied matrix models for 2D quantum gravity and string theory [28, 53]. At the same time, it generalizes it, as we are going to explain, by incorporating the insights coming from canonical loop quantum gravity and its covariant spin foam formulation of the dynamics, and so it became an important part of this approach to the quantization of 4D gravity [72, 74, 81, 85]. Furthermore, it is a point of convergence of the same loop quantum gravity approach and of simplicial quantum gravity approaches, like quantum Regge calculus [93] and dynamical triangulations [3, 79], in that the covariant dynamics of the first takes the form, as we are going to see, of simplicial path integrals.