Given a marked ∞-category D† (i.e. an ∞-category equipped with a specified collection of morphisms) and a functor F: D→ B with values in an ∞-bicategory, we define [InlineEquation not available: see fulltext.], the marked colimit of F. We provide a definition of weighted colimits in ∞-bicategories when the indexing diagram is an ∞-category and show that they can be computed in terms of marked colimits. In the maximally marked case D♯, our construction retrieves the ∞-categorical colimit of F in the underlying ∞-category B⊆ B. In the specific case when [InlineEquation not available: see fulltext.], the ∞-bicategory of ∞-categories and D♭ is minimally marked, we recover the definition of lax colimit of Gepner–Haugseng–Nikolaus. We show that a suitable ∞-localization of the associated coCartesian fibration Un D(F) computes [InlineEquation not available: see fulltext.]. Our main theorem is a characterization of those functors of marked ∞-categories f: C†→ D† which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits.