  Given a newform f, we extend Howard's results on the variation of Heegner
points in the Hida family of f to a general quaternionic setting. More
precisely, we build big Heegner points and big Heegner classes in terms of
compatible families of Heegner points on towers of Shimura curves. The novelty
of our approach, which systematically exploits the theory of optimal
embeddings, consists in treating both the case of definite quaternion algebras
and the case of indefinite quaternion algebras in a uniform way. We prove
results on the size of Nekov\'a\v{r}'s extended Selmer groups attached to
suitable big Galois representations and we formulate two-variable Iwasawa main
conjectures both in the definite case and in the indefinite case. Moreover, in
the definite case we propose refined conjectures \`a la Greenberg on the
vanishing at the critical points of (twists of) the L-functions of the modular
forms in the Hida family of f living on the same branch as f.
