  Homotopy limits and colimits are homotopical replacements for the usual
limits and colimits of category theory, which can be approached either using
classical explicit constructions or the modern abstract machinery of derived
functors. Our first goal in this paper is expository: we explain both
approaches and a proof of their equivalence. Our second goal is to generalize
this result to enriched categories and homotopy weighted limits, showing that
the classical explicit constructions still give the right answer in the
abstract sense. This result partially bridges the gap between classical
homotopy theory and modern abstract homotopy theory. To do this we introduce a
notion of "enriched homotopical categories", which are more general than
enriched model categories, but are still a good place to do enriched homotopy
theory. This demonstrates that the presence of enrichment often simplifies
rather than complicates matters, and goes some way toward achieving a better
understanding of "the role of homotopy in homotopy theory."
