  We investigate general Shapiro-Lopatinsky elliptic boundary value problems on
manifolds with polycylindrical ends. This is accomplished by compactifying such
a manifold to a manifold with corners of in general higher codimension, and we
then deal with boundary value problems for cusp differential operators. We
introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary
value problems, and construct parametrices for elliptic cusp operators within
this calculus. Fredholm solvability and elliptic regularity up to the boundary
and up to infinity for boundary value problems on manifolds with
polycylindrical ends follows.
