  Spinfoams provide a framework for the dynamics of loop quantum gravity that
is manifestly covariant under the full four-dimensional diffeomorphism symmetry
group of general relativity. In this way they complete the ideal of
three-dimensional diffeomorphism covariance that consistently motivates loop
quantum gravity at every step. Specifically, spinfoam models aim to provide a
projector onto, and a physical inner product on, the simultaneous kernel of all
of the constraints of loop quantum gravity by means of a discretization of the
gravitational path integral. In the limit of small Planck constant, they are
closely related to the path integral for Regge calculus, while at the same time
retaining all of the tools of a canonical quantum theory of gravity. They may
also be understood as generalizations of well-understood state sum models for
topological quantum field theories. In this chapter, we review all of these
aspects of spinfoams, as well as review in detail the derivation of the
currently most used spinfoam model, the EPRL model, calculational tools for it,
and the various extensions of it in the literature. We additionally summarize
some of the successes and open problems in the field.
