Manifolds are, of course, by definition spaces which are locally Euclidean space, so the term "nonlocally linear manifold" is an oxymoron. Nontheless there is a fascinating class of spaces which deserve that title in that they seem likely to possess many of the geometric properties of manifolds and some of the analytic ones. Moreover, these spaces fill in seeming gaps in the theory of manifolds.
These spaces (constructed by Bryant, Ferry, Mio, and speaker) are the "ANR homology manifolds which satisfy the disjoint disk condition" (the latter is internal two-dimensional transversality) were indeed conjectured (largely because of work of Edwards and Cannon) to be manifolds. However, the formal structure of high dimensional geometrical topology is greatly enhanced by the inclusion of these spaces. (Thus, the theory gains periodicities and functorialities that conventional manifold theory does not have.)
There is a beautiful analogy between the theory of these spaces and the classification theory of group actions on manifolds which are not assumed to be smooth locally, but instead are only demanded to have locally flat fixed point sets. Many ideas involving the analysis ($G$-signature theorems and signautre operators) and geometry of such actions suggest results and conjectures about homology manifolds. On the other hand, the conjectured local structure of homology manifolds, and known results about their classification up to $s$-cobordism gives these objects a greater algebraic simplicity than the orbifold case, and has also led to new insights in transformation groups.
The talk itself will deal with: