  Unifying several directions of the development of the study of summing
multilinear operators between Banach spaces, we construct a general framework
that studies, under one single definition, multilinear operators that are
summing with respect to sums taken over any number of indices, iterated and
non-iterated sums (the isotropic and the anisotropic cases), sums over
arbitrary blocks and over several different sequence norms. A large number of
special classes of multilinear operators and of methods of generating classes
of multilinear operators are recovered as particular instances. Ideal
properties and coincidence theorems for the general classes are proved.
