We first discuss Brown's category of fibrant objects structures on closed monoidal categories by means of some specific arrows called pseudo-cofibrations. These arrows are the ones whose pullback-power with (acyclic) fibrations are also (acyclic) fibrations; which can be defined whenever the underlying category is closed monoidal or enriched over a category of fibrant objects. Later we discuss the category of fibrant objects structures on enriched categories. If V is a closed monoidal category with a category of fibrant object structure on it and C is enriched over V and powered over a colimit dense subcategory of V, then under mild conditions C can also be made into a category of fibrant objects. We discuss constructions and properties of these structures and their extension to the equivariant setting. Lastly, we give some already existing and some new examples of such categories of fibrant objects and mention some applications.

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