  We review some approaches to the Hamiltonian dynamics of (loop) quantum
gravity, the main issues being the regularization of the Hamiltonian and the
continuum limit. First, Thiemann's definition of the quantum Hamiltonian is
presented, and then more recent approaches. They are based on toy models which
provide new insights into the difficulties and ambiguities faced in Thiemann's
construction. The models we use are parametrized field theories, the
topological BF model of which a special case is three-dimensional gravity which
describes quantum flat space, and Regge lattice gravity.
