  We systematically calculate the radiative corrections of order $\alpha/\pi$
to elastic muon-proton scattering at low momentum transfers. These include
vacuum polarization, photon-loop form factors of the muon and the proton,
two-photon exchange corrections and soft photon radiation. In particular, we
discuss these corrections for the kinematics of the upcomimg AMBER experiment
with a $100\,$GeV muon beam energy. It is found that for the ratio to the Born
cross section, only the minor terms from the photon-loop form factors of the
proton and two-photon exchange depend on the proton structure predetermined by
the strong interactions. Since a prominent role among the radiative corrections
is played by soft photon radiation, the calculation of the bremsstrahlung
process $\mu p \to\mu p \gamma$ should be extended beyond the soft photon
approximation and tailored to the specific experimental conditions.
