  To match the precision of present and future measurements of W-boson
production at hadron colliders electroweak radiative corrections must be
included in the theory predictions. In this paper we consider their effect on
the transverse momentum (p_T) distribution of W bosons, with emphasis on large
p_T. We evaluate the full electroweak O(alpha) corrections to the processes pp
-> W+jet and p\bar p -> W+jet including virtual and real photonic
contributions. We present the explicit expressions in analytical form for the
virtual corrections and provide results for the real corrections, discussing in
detail the treatment of soft and collinear singularities. We also provide
compact approximate expressions which are valid in the high-energy region,
where the electroweak corrections are strongly enhanced by logarithms of
\hat{s}/M_W^2. These expressions describe the complete asymptotic behaviour at
one loop as well as the leading and next-to-leading logarithms at two loops.
Numerical results are presented for proton-proton collisions at 14 TeV and
proton-antiproton collisions at 2 TeV. The corrections are negative and their
size increases with p_T. At the LHC, where transverse momenta of 2 TeV or more
can be reached, the one- and two-loop corrections amount up to -40% and +10%,
respectively, and will be important for a precise analysis of W production. At
the Tevatron, transverse momenta up to 300 GeV are within reach. In this case
the electroweak corrections amount up to -10% and are thus larger than the
expected statistical error.
