  $Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some
aspects of local differential geometry are formulated as $Vect(N)$
representation theory. There is a new class of modules, {\it conformal fields},
whose restrictions to the subalgebra $sl(N+1) \subset Vect(N)$ are
finite-dimensional $sl(N+1)$ representations. In this regard they are simpler
than tensor fields. Fock modules are also constructed. Infinities, which are
unremovable even by normal ordering, arise unless bosonic and fermionic degrees
of freedom match.
