In this paper a family of one-dimensional nonlinear systems which model the blood pulse propagation in compliant arteries is presented and investigated. They are obtained by averaging the Navier-Stokes equation on each section of an arterial vessel and using simplified models for the vessel compliance. Different differential operators arise depending on the simplifications made on the structural model. Starting from the most basic assumption of pure elastic instantaneous equilibrium, which provides a well-known algebraic relation between intramural pressure and vessel section area, we analyse in turn the effects of terms accounting for inertia, longitudinal prestress and viscoelasticity. The problem of how to account for branching and possible discontinuous wall properties is addressed, the latter aspect being relevant in the presence of prosthesis and stents. To this purpose a domain decomposition approach is adopted and the conditions which ensure the stability of the coupling are provided. The numerical method here used in order to carry out several test cases for the assessment of the proposed models is based on a finite element Taylor-Galerkin scheme combined with operator splitting techniques