Let \(S\) and \(T\) be compact sets, \(C(S)\) and \(C(T)\) Banach spaces of real continuous functions on \(S\) and \(T\), respectively, \(\varphi: S\to T\) a continuous mapping, and let \(\varphi^0: C(T)\to C(S)\) be defined by \(\varphi^0g= g\circ\varphi\), \(g\in C(T)\). A sublinear operator \(P: C(S)\to C(T)\) is called a sublinear exave for \(\varphi\) if \[ \varphi^0 P\varphi^0= \varphi^0. \] The class of sublinear exaves contains sublinear extension and averaging operators. The class of linear exaves was introduced by A. Pełczyński. General properties of such sublinear exaves are studied. An integral representation is obtained. An existence theorem for a sublinear averaging operator is proved. Connections of exaves with continuous selections are examined. Applications to the theory of sublinear operators, multivalued mappings, and linear exaves are indicated.