  Spin foam vertex amplitudes are the key ingredient of spin foam models for
quantum gravity. These fall into the realm of discretized path integral, and
can be seen as generalized lattice gauge theories. They can be seen as an
attempt at a 4-dimensional generalization of the Ponzano-Regge model for 3d
quantum gravity. We motivate and review the construction of the vertex
amplitudes of recent spin foam models, giving two different and complementary
perspectives of this construction. The first proceeds by extracting geometric
configurations from a topological theory of the BF type, and can be seen to be
in the tradition of the work of Barrett, Crane, Freidel and Krasnov. The second
keeps closer contact to the structure of Loop Quantum Gravity and tries to
identify an appropriate set of constraints to define a Lorentz-invariant
interaction of its quanta of space. This approach is in the tradition of the
work of Smolin, Markopoulous, Engle, Pereira, Rovelli and Livine.
