Abstract An axiparabola is a reflective aspherical optics that focuses a light beam into an extended focal line. The light intensity and group velocity profiles along the focus are adjustable through the proper design. The on-axis light velocity can be controlled, for instance, by adding spatio-temporal couplings via chromatic optics on the incoming beam. Therefore the energy deposition along the axis can be either subluminal or superluminal as required in various applications. This article first explores how the axiparabola design defines its properties in the geometric optics approximation. Then the obtained description is considered in numerical simulations for two cases of interest for laser-plasma acceleration. We show that the axiparabola can be used either to generate a plasma waveguide to overcome diffraction or for driving a dephasingless wakefield accelerator.