  We propose multidimensional versions of the Painlev\'e VI equation and its
degenerations. These field theories are related to the isomonodromy problems of
flat holomorphic infinite rank bundles over elliptic curves and take the form
of non-autonomous Hamiltonian equations. The modular parameter of curves plays
the role of "time". Reduction of the field equations to the zero modes leads to
${\rm SL}(N, {\mathbb C})$ monodromy preserving equations. The latter coincide
with the Painlev\'e VI equation for $N\!=\!2$. We consider two types of the
bundles. In the first one the group of automorphisms is the centrally and
cocentrally extended loop group $L({\rm SL}(N, {\mathbb C}))$ or some multiloop
group. In the case of the Painlev\'e VI field theory in D=1+1 four constants of
the Painlev\'e VI equation become dynamical fields. The second type of bundles
are defined by the group of automorphisms of the noncommutative torus. They
lead to the equations in dimension 2+1. In both cases we consider
trigonometric, rational and scaling limits of the theories. Generically (except
some degenerate cases) the derived equations are nonlocal. We consider Whitham
quasiclassical limit to integrable systems. In this way we derive two and three
dimensional integrable nonlocal versions of the integrable Euler-Arnold tops.
