  The excess of high-$Q^2$ events found by H1 and ZEUS at HERA in $e^+p$
deep-inelastic scattering above the Standard Model prediction motivates us to
calculate the NLO QCD corrections to the HERA scalar leptoquark (or squark)
production cross sections. We find that the corrections are significant, of
order 50% in the mass range of interest. We also calculate the leptoquark
average transverse momentum squared and find it to be rather small. Various
leptoquark production cross sections at the Tevatron $p \bar p$ collider are
also considered. We investigate in detail the leptoquark interpretation of the
HERA data. First we assume a minimal leptoquark model with a single diagonal
Yukawa coupling to first family lepton and quark mass eigenstates only. In this
case constraints from atomic parity violating experiments allow only isodoublet
scalar leptoquark production at HERA. This interpretation can be confirmed or
ruled out in the near future by high luminosity data at the Tevatron. The
Tevatron data already appear to rule out the vector leptoquark interpretation
of the HERA data. We also consider a more general model which allows for all
possible left-handed, right-handed, flavour and lepton number changing
couplings. The allowed values of the Yukawa couplings of this general model
offer several different interpretations of the data which are radically
different from the minimal model solutions. However these somewhat ad hoc tuned
solutions can easily be tested by future HERA experiments.
