As they are defined using structure sheaves, locally definable spaces and more sophisticated weakly definable spaces are a natural generalization of both the subanalytic and the definable o- minimal settings. If considered over an o-minimal expansion of a field, they have a well developed homotopy theory, which in particular gives different kinds of homology and cohomology. The recommended reading for my lecture is the preprint arXiv:0904.4896 (http://arxiv.org/abs/0904.4896).