An -category is the special case of -category for .
It is best known now through a geometric definition of higher category.
the definition by Carlos Simpson and Tamsamani;
the definition in terms of n-fold Segal spaces;
a definition in terms of scaled simplicial sets, following Verity’s simplicial model for weak omega-categories by Jacob Lurie (see reference below)
See also the list of all definitions of higher categories at (∞,n)-category.
In (∞,2)-Categories and the Goodwillie Calculus Jacob Lurie discusses a variety of model category structures, all Quillen equivalent, that all model the (∞,1)-category of -categories, in generalization of the standard model category models for (∞,1)-categories themselves (see there for details).
between Joyal--enriched categories, Joyal--enriched complete Segal spaces and simplicial Joyal-simplicial sets.
This is remark 0.0.4, page 5 of the article. There are many more models. See there for more.
Classes of examples include
For a suitable monoidal (∞,1)-category there is the -category of -algebras and -bimodules in . See at bimodule - Properties - The (∞,2)-category of ∞-algebras and ∞-bimodules.