  We develop a general framework (multidimensional asymptotic classes, or
m.a.c.s) for handling classes of finite first order structures with a strong
uniformity condition on cardinalities of definable sets: The condition asserts
that definable families given by a formula \phi(x,y) should take on a fixed
number n_\phi of approximate sizes in any M in the class, with those sizes
varying with M. The prototype is the class of all finite fields, where the
uniformity is given by a theorem of Chatzidakis, van den Dries and Macintyre.
It inspired the development of asymptotic classes of finite structures, which
this new framework extends.
  The underlying theory of m.a.c.s is developed, including preservation under
bi-interpretability, and a proof that for the m.a.c. condition to hold it
suffices to consider formulas \phi(x,y) with x a single variable. Many examples
of m.a.c.s are given, including 2-sorted structures (F,V) where V is a vector
space over a finite field F possibly equipped with a bilinear form, and an
example arising from representations of quivers of finite representation type.
We also give examples and structural results for multidimensional exact classes
(m.e.c.s), where the definable sets take a fixed number of precisely specified
cardinalities, which again vary with M.
  We also develop a notion of infinite generalised measurable structure,
whereby definable sets are assigned values in an ordered semiring. We show that
any infinite ultraproduct of a m.a.c. is generalised measurable, that values
can be taken in an ordered ring if the m.a.c. is a m.e.c., and explore
model-theoretic consequences of generalised measurability. Such a structure
cannot have the strict order property, and stability-theoretic properties can
be read off from the measures in the semiring.
