We report on experiments that were performed with microwave waveguide systems and demonstrate that in the frequency range of a single transversal mode they may serve as a model for closed and open quantum graphs. These consist of bonds that are connected at vertices. On the bonds, they are governed by the one-dimensional Schrödinger equation with boundary conditions imposed at the vertices. The resulting transport properties through the vertices may be expressed in terms of a vertex scattering matrix. Quantum graphs with incommensurate bond lengths attracted interest within the field of quantum chaos because, depending on the characteristics of the vertex scattering matrix, its wave dynamic may exhibit features of a typical quantum system with chaotic counterpart. In distinction to microwave networks, which serve as an experimental model of quantum graphs with Neumann boundary conditions, the vertex scattering matrices associated with a waveguide system depend on the wave number and the wave functions can be determined experimentally. We analyze the spectral properties of microwave waveguide systems with preserved and partially violated time-reversal invariance, and the properties of the associated wave functions. Furthermore, we study properties of the scattering matrix describing the measurement process within the framework of random matrix theory for quantum chaotic scattering systems.