  This chapter, written for "Stable categories and structured ring spectra,"
edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the
history of homotopical categories, from Gabriel and Zisman's categories of
fractions to Quillen's model categories, through Dwyer and Kan's simplicial
localizations and culminating in $(\infty,1)$-categories, first introduced
through concrete models and later re-conceptualized in a model-independent
framework. This reader is not presumed to have prior acquaintance with any of
these concepts. Suggested exercises are included to fertilize intuitions and
copious references point to external sources with more details. A running theme
of homotopy limits and colimits is included to explain the kinds of problems
homotopical categories are designed to solve as well as technical approaches to
these problems.

Algebraic Topology;Category Theory