  We perform an exhaustive analysis of the Equivalence Theorem both in the
minimal Standard Model and in an Effective Electroweak Chiral Lagrangian up to
${\cal O}(p^4)$. We have considered the leading corrections to the usual
prescription consisting in just replacing longitudinally polarized $W$ or $Z$
by the corresponding Goldstone bosons. The corrections appear through an
overall constant multiplying the Goldstone boson amplitude as well as through
additional diagrams. By including them we can extend the domain of
applicability of the Equivalence Theorem, making it suitable for precision
tests of the symmetry breaking sector of the Standard Model. The on-shell
scheme has been used throughout. When considering the Equivalence Theorem in an
Effective Chiral lagrangian we analyze its domain of applicability, as well as
several side issues concerning gauge fixing, Ward identities, on-shell scheme
and matching conditions in the effective theory. We have analyzed in detail the
processes $W^+ W^- \to W^+W^-$ and $W^+W^+\to W^+W^+$ to illustrate the points
made.
