We present a geometric construction of Backlund transformations and
discretizations for a large class of algebraic completely integrable systems.
To be more precise, we construct families of Backlund transformations, which
are naturally parametrized by the points on the spectral curve(s) of the
system. The key idea is that a point on the curve determines, through the
Abel-Jacobi map, a vector on its Jacobian which determines a translation on the
corresponding level set of the integrals (the generic level set of an algebraic
completely integrable systems has a group structure). Globalizing this
construction we find (possibly multi-valued, as is very common for Backlund
transformations) maps which preserve the integrals of the system, they map
solutions to solutions and they are symplectic maps (or, more generally,
Poisson maps). We show that these have the spectrality property, a property of
Backlund transformations that was recently introduced. Moreover, we recover
Backlund transformations and discretizations which have up to now been
constructed by ad-hoc methods, and we find Backlund transformations and
discretizations for other integrable systems. We also introduce another
approach, using pairs of normalizations of eigenvectors of Lax operators and we
explain how our two methods are related through the method of separation of
variables.