We introduce open quasitoric manifolds and their functorial properties, including complex bundle maps of their stable tangent bundles, and relate these new spaces to the standard constructions of toric topology: quasitoric manifolds, moment angle manifolds and polyhedral products. We extend the domain of these constructions to countably infinite simplicial complexes, clarifying and generalising constructions of Davis and Januszkiewicz. In particular we describe final spaces in the categories of open quasitoric manifolds and quasitoric spaces, as well as in the categories of characteristic pairs and dicharacteristic pairs. We show how quasitoric manifolds can be constructed smoothly as pullbacks of the final spaces QT(n) for n >= 1, and how stably complex structure also arises this way. We calculate the integral cohomology of quasitoric spaces over Cohen-Macaulay simplicial complexes, including the final spaces QT(n) as a special case. We describe a procedure for calculating the Chern numbers of a quasitoric manifold M and, relating this to our cohomology calculations, show how it may be interpreted in terms of the simplicial homology of H(n), the simplicial complex underlying QT(n).