  In a previous work, we have introduced a weakening of Quillen model
categories called weak model categories. They still allow all the usual
constructions of model category theory, but are easier to construct and are in
some sense better behaved. In this paper we continue to develop their general
theory by introducing combinatorial and accessible weak model categories. We
give simple necessary and sufficient conditions under which such a weak model
category can be extended into a left and/or right semi-model category. As an
application, we recover Cisinski-Olschok theory and generalize it to weak and
semi-model categories. We also provide general existence theorems for both left
and right Bousfield localization of combinatorial and accessible weak model
structures, which combined with the results above gives existence results for
left and right Bousfield localization of combinatorial and accessible left and
right semi-model categories, generalizing previous results of Barwick.
Surprisingly,we show that any left or right Bousfield localization of an
accessible or combinatorial Quillen model category always exists, without
properness assumptions, and is simultaneously both a left and a right
semi-model category, without necessarily being a Quillen model category itself.
