  We study several notions of shift equivalence for C*-correspondences and the
effect that these equivalences have on the corresponding Pimsner dilations.
Among others, we prove that non-degenerate, regular, full C*-correspondences
which are shift equivalent have strong Morita equivalent Pimsner dilations. We
also establish that the converse may not be true. These results settle open
problems in the literature.
  In the context of C*-algebras, we prove that if two non-degenerate, regular,
full C*-correspondences are shift equivalent, then their corresponding
Cuntz-Pimsner algebras are strong Morita equivalent. This generalizes results
of Cuntz and Krieger and Muhly, Tomforde and Pask. As a consequence, if two
subshifts of finite type are eventually conjugate, then their Cuntz-Krieger
algebras are strong Morita equivalent.
  Our results suggest a natural analogue of the Shift Equivalence Problem in
the context of C*-correspondences. Even though we do not resolve the general
Shift Equivalence Problem, we obtain a positive answer for the class of
imprimitivity bimodules.
