  We give an overview of the generalized Calder\'on-Zygmund theory for
"non-integral" singular operators, that is, operators without kernels bounds
but appropriate off-diagonal estimates. This theory is powerful enough to
obtain weighted estimates for such operators and their commutators with $\BMO$
functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important
role and we present them. They are particularly well suited to the semigroups
generated by second order elliptic operators and the range of exponents $(p,q)$
rules the $L^p$ theory for many operators constructed from the semigroup and
its gradient. Such applications are summarized.
