Boundaries play an important role for pattern formation in hydrodynamic systems, like in e.g. in Taylor-Couette flow. This is the flow of a viscous liquid in the gap between two concentric rotating cylinders. In experimental systems up- and downwards propagating spiral vortices are often observed as primary pattern from an oscillatory bifurcation for sufficiently high rate of counter-rotation. The onset of spirals in experiments has been interpreted from models of a symmetry-breaking Hopf-bifurcation under the assumption of an O(2)-symmetry, i.e. an axial translational and reflection symmetry, of the flow. Since experimental flows are always of finite axial extend the translational symmetry is broken. Theory of Hopf-bifurcation with broken translational symmetry (Knobloch, Danglmayr, Pierce, Hirschberg, Landsberg) reveals a more complicated bifurcation structure including standing waves, modulated waves, homoclinic bifurcations, Takens-Bogdanov points and chaos. We show that these essential elements of the theory are robust and that the theory is appropriate to describe the onset of spiral vortices in the Taylor-Couette experiment. The effect of system length is discussed. In a Taylor-Couette experiment with an additional axial Poiseuille flow the reflection symmetry is broken. Our experimental results on the onset of spiral vortices in a so-called Spiral-Poiseuille flow with an aspect ratio larger than 20 reveal a general agreement with theoretical predictions from Hopf-bifurcation with broken reflection symmetry (Crawford, Knobloch). Aspects of the bifurcation structure related to convective instabilities as well as coupling of the mean flow to nonlinear spirals are also discussed.