  In this work we use the bigonal, trigonal and tetragonal constructions to
describe the fibers of the Prym map P : R_{g} ---->A_{g-1} inthe cases when it
is dominant, i.e. for g < 7. The most interesting cases are g = 5, where the
fiber is a double cover of the Fano surface of lines on a cubic threefold, and
g=6, where the map is generically finite (of degree 27) with Galois group
WE_{6}, so that the general fiber has the structure of the 27 lines on a cubic
surface. For g > 6, the map is known to be generically injective. The
tetragonal construction gives many counterexamples to injectivity, and we
conjecture that all noninjectivity is due to the tetragonal construction.
