  We carry over to a quite general noncommutative setting some of the basic
tools of differential geometry, using from the very beginning the setting of
convenient vector spaces developed by Froelicher and Kriegl, which allows to
carry all of multilinear algebra into this kind of functional analysis with
suitably completed tensor products. In the first section we give a short
description of the setting of convenient spaces elaborating those aspects which
are needed later. Then we repeat the usual construction of noncommutative
differential forms for convenient algebras. Next they show that the bimodule
$\Omega\sb n (A)$ of universal non-commutative $n$-forms represents the functor
of the normalized Hochschild $n$-cocycles. In the third section we introduce
the noncommutative version of the Froelicher-Nijenhuis bracket by investigating
all bounded graded derivations of the algebra of differential forms. This
bracket is then used to formulate the concept of integrability and
involutiveness for distributions and to indicate a route towards a theorem of
Frobenius. This is then used to discuss bundles and connections in the
noncommutative setting and to go some steps towards a noncommutative Chern-Weil
homomorphism. In the final section we give a brief description of the
noncommutative version of the Schouten-Nijenhuis bracket and describe Poisson
structures.
