This is a sequel to an earlier paper by the author [ibid. 111, No. 4, 613-665 (1987; Zbl 0627.58034)]. There, he introduced and developed the finite-dimensional representation theory of a particular generalization of the concept of a compact Lie group, which has the desirable property of admitting nontrivial deformations. (A more abstract schema has been proposed to the same end by \textit{V. G. Drinfel'd} [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)]. Here the author discusses the foundations of (noncommutative) differential geometry for these objects. A ``differential calculus'' over an algebra \(\mathcal A\) (generalizing the algebra of smooth functions) consists, in principle, of an \(\mathcal A\)- bimodule \(\Gamma\) (generalizing the module of 1-forms), and a derivation \(d:{\mathcal A}\to \Gamma\) (corresponding to the exterior derivative). For the author's pseudogroups, the algebra \(\mathcal A\) was part of the definition; however, the choice of differential calculus over \(\mathcal A\) is not canonical, and he remarks that nonstandard calculi may be constructed even on classical compact groups. Given a differential calculus over \(\mathcal A\) which satisfies appropriate covariance conditions, it is possible --- with effort and ingenuity --- to construct suitable, not always transparent, analogues of tensors and forms, of the exterior derivative in all degrees, of left-invariant vector fields and their brackets, of the Maurer-Cartan equation, and of the Jacobi identity. The author comments that for these purposes the ``compactness'' of his pseudogroups, meaning certain algebraic identities of orthogonality in the definition, does not appear essential.