  We implement scalar and vector leptoquark (LQ) models in the universal
FeynRules output (UFO) format assuming the Standard Model fermion content and
conservation of baryon and lepton numbers. Scalar LQ implementations include
next-to-leading order (NLO) QCD corrections. We report the NLO QCD inclusive
cross sections in proton-proton collisions at 13 TeV, 14 TeV, and 27 TeV for
all on-shell LQ production processes. These comprise (i) LQ pair production ($p
p \to \Phi \Phi$) and (ii) single LQ + lepton production ($p p \to \Phi \ell$)
for all initial quark flavours ($u$, $d$, $s$, $c$, and $b$). Vector LQ
implementation includes adjustable non-minimal QCD coupling. We discuss several
aspects of LQ searches at a hadron collider, emphasising the implications of
$SU(2)$ gauge invariance, electroweak and flavour constraints, on the possible
signatures. Finally, we outline the high-$p_T$ search strategy for LQs recently
proposed in the literature to resolve experimental anomalies in $B$-meson
decays. In this context, we stress the importance of complementarity of the
three LQ related processes, namely, $p p \to \Phi \Phi$, $p p \to \Phi \ell$,
and $p p \to \ell \ell$.
