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Multipactor discharges in waveguides with wedge- shaped cross section are studied both analytically and by means of numerical simulations. It is shown that similar to the case of a rectangular waveguide, a trapped electron trajectory is possible in a wedge-shaped waveguide; however, this trajectory is unstable with respect to small deviations of launch time and initial position of the electrons. Numerical simulations have been carried out taking into account a spread of the electron initial velocity and the action of the radio frequency magnetic field on the electron motion. The simulation results show that growth of the multipactor avalanche is possible in the wedge-shaped waveguide; however, the multipactor threshold in terms of the transmitted power is higher than for the rectangular waveguide with the same propagation constant and cross-section area.