In this paper we study the interaction of a shock with an axisymmetric longitudinal vortex. A linearized analysis for small vortex strength is performed, and compared with results from a high-order axisymmetric shock-fitted Euler solution. It is confirmed that for weak vortices, predictions from linear theory agree well with results from nonlinear numerical simulations at the shock location. To handle very strong longitudinal vortices, which may ultimately break the shock, we use an axisymmetric high-order essentially non-oscillatory (ENO) shock-capturing scheme. Comparisons of shock-captured and shock-fitted results are performed in their regions of common validity. We also study the vortex breakdown as a function of Mach number ranging from 1.3 to 10, thus extending the range of existing results. For vortex strengths above a critical value, a triple point forms on the shock, leading to a Mach disk. This leads to a strong recirculating region downstream of the shock and a secondary shock forms to provide the necessary deceleration so that the fluid velocity can adjust to downstream subsonic conditions.