  Cubical type theory provides a constructive justification to certain aspects
of homotopy type theory such as Voevodsky's univalence axiom. This makes many
extensionality principles, like function and propositional extensionality,
directly provable in the theory. This paper describes a constructive semantics,
expressed in a presheaf topos with suitable structure inspired by cubical sets,
of some higher inductive types. It also extends cubical type theory by a syntax
for the higher inductive types of spheres, torus, suspensions,truncations, and
pushouts. All of these types are justified by the semantics and have judgmental
computation rules for all constructors, including the higher dimensional ones,
and the universes are closed under these type formers.
