The bounded version of the eight-vertex model of Statistical Mechanics is investigated. We study square, diamond and general finite domains on the square lattice and give exact characterizations to legal boundary conditions and number of fill-ins. The sets of legal configurations with a given boundary turn out always to have the graph topology of a hypercube with a particularly simple edge action. This enables a simple probabilistic description of the configurations as well as an efficient configuration generation using a cellular automaton. Finally, by invoking height functions we study restricted edge action which leads to ice-model as well as to lesser know vertex models, some subsets of the eight-vertex model, some not.