Teichmueller curves are curves embedded in the moduli spaces of curves which are geodesic for the Teichmueller metric. Their universal covers naturally live in Teichmueller space and can be seen there as complex geodesics. Such objects arose naturally in the wake of Thurston's work on diffeomorphisms of surfaces and their study (initiated by H.Masur, W.Veech and others in the late seventies) a priori pertains more to the field of dynamical systems, especially ergodic theory. It was however recognized a few years ago that these curves also have very rich (and largely unexplored to-date) algebraic and arithmetic properties. I will present (a small part of) this vast landscape and then focus on some particular examples like the family of `origamis' which on the one hand can claim to represent a higher dimensional analog of `dessins d'enfants' (with a richer geometric setting) and on the other hand are closely related to `Hurwitz curves'.