  A unified treatment of both chiral and radiative corrections to the
low-energy elastic lepton-proton scattering processes is presented in Heavy
Baryon Chiral Perturbations Theory. The proton hadronic chiral corrections
include the next-to-next-to leading order corrections whereas the radiative
corrections include the next-to-leading order terms in our novel power counting
scheme. We find that the net fractional well-defined chiral corrections with
respect to the leading order Born cross section can be as large as $10\%$
($20\%$) for electron (muon) scattering process for MUon proton Scattering
Experiment (MUSE) kinematics. We show {\it via} our model-independent treatment
of the low-energy lepton-proton kinematics, that the largest theoretical
uncertainty is due to the recent different published values of the proton's rms
radius while, e.g., the next higher order hadronic chiral terms are expected to
give rather nominal errors. For the radiative corrections we demonstrate a
systematic order by order cancellation of all infrared singularities and
present our finite ultraviolet regularization results. We find that the
radiative corrections for muon-proton scattering is of the order of $2\%$,
whereas for electron scattering the radiative corrections could be as large as
$25\%$. We attribute such a contrasting result partially to the fact that in
muon scattering the leading radiative order correction goes through zero in
some intermediate low-momentum transfer region, leaving the sub-leading
radiative chiral order effects to play a dominant role in this particular
kinematic region. For the low-energy MUSE experiment, the often neglected
lepton mass as well as the Pauli form factor contributions of the relativistic
leptons are incorporated in all our computations.
