Quasitoric spaces were introduced by Davis and Januskiewicz
in their 1991 Duke paper. There they extensively studied topological
invariants of quasitoric manifolds. These manifolds are generalizations
or topological counterparts of nonsingular projective toric
varieties. In this article we study structures and invariants
of quasitoric orbifolds. In particular, we discuss equivalent
definitions and determine the orbifold fundamental group,
rational homology groups and cohomology ring of a quasitoric
orbifold. We determine whether any quasitoric orbifold can
be the quotient of a smooth manifold by a finite group action
or not. We prove existence of stable almost complex structure
and describe the Chen--Ruan cohomology groups of an almost
complex quasitoric orbifold.