We explore interpretations of the anomaly observed by H1 and ZEUS at HERA in deep-inelastic e + p scattering at very large Q 2, in terms of possible physics beyond the Standard Model. Since the present data could be compatible with either a continuum or a resonant solution, we discuss both the possibilities of new effective interactions and the production of a narrow state of mass M ∼ 200 GeV with leptoquark couplings. We compare these models with the measured Q 2 distributions: for the contact terms, constraints from LEP 2 and the Tevatron allow only a few choices of helicity and flavour structure that could roughly fit the HERA data. The data are instead quite consistent with the Q 2 distribution expected from a leptoquark state. We study the production cross sections of such a particle at the Tevatron and at HERA, the latter in the cases where it is produced from either a valence or a sea quark. The absence of a signal at the Tevatron disfavours the likelihood that any such leptoquark decays only into e + q. We then focus on the possibility that the leptoquark is a squark with R-violating couplings. In view of the present experimental limits on such couplings, the most likely production channels are e +d → c ̃ L or perhaps e +d → t ̃ , with e +s → t ̃ a more marginal possibility. We point out that the c ̃ L could have competing branching ratios for R-violating decay channels, whereas t ̃ decays would be more likely to be dominated by one or the other. Possible tests of our preferred model include the absence both of analogous events in e − p collisions and of charged current events, and the presence of detectable cascade decays whose kinematical signatures we discuss. This model could also make an observable contribution to K → π v − v and/or neutrinoless ββ decay. We also discuss the possible implications for the Tevatron and for e +e − → q − q and neutralinos at LEP 2.