  We give the definitions of model bicategory and $q$-homotopy, which are
natural generalizations of the notions of model category and homotopy to the
context of bicategories. For any model bicategory $\mathcal{C}$, denote by
$\mathcal{C}_{fc}$ the full sub-bicategory of the fibrant-cofibrant objects. We
prove that the 2-dimensional localization of $\mathcal{C}$ at the weak
equivalences can be computed as a bicategory $\mathcal{H}o(\mathcal{C})$ whose
objects and arrows are those of $\mathcal{C}_{fc}$ and whose 2-cells are
classes of $q$-homotopies up to an equivalence relation. When considered for a
model category, $q$-homotopies coincide with the homotopies as considered by
Quillen. The pseudofunctor $\mathcal{C} \stackrel{q}{\longrightarrow}
\mathcal{H}o(\mathcal{C})$ which yields the localization is constructed by
using a notion of fibrant-cofibrant replacement in this context. We include an
appendix with a general result of independent interest on a transfer of
structure for lax functors, that we apply to obtain a pseudofunctor structure
for the fibrant-cofibrant replacement.
