The Teichmüller space of a compact 2-orbifold X can be defined as the space of faithful and discrete representations of the fundamental group $\pi_1$(X) of X into PGL(2,R). It is a contractible space. For closed orientable surfaces, "higher analogues" of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of $\pi_1$(X) that consist entirely of discrete and faithful representations. There are two known families of such spaces, namely Hitchin representations and maximal representations, and conjectures on how to find others. In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yields new examples of higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.