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Modifications to polar-gap models for pulsars are discussed for the casewhere the surface magnetic field, $B_\S$, of the neutron star is strong. For$B\ga4\times10^8\rm\,T$, the curvature $\gamma$-quanta emitted tangentially tothe curved force lines of the magnetic field are captured near the threshold ofbound pair creation and are channelled along the magnetic field as boundelectron-positron pairs (positronium). The stability of such bound pairsagainst ionization by the parallel electric field, $E_\parallel$, in the polarcap, and against photoionization is discussed. Unlike free pairs, bound pairsdo not screen $E_\parallel$ near the neutron star. As a consequence, the energyflux in highly relativistic particles and high-frequency (X-ray and/or$\gamma$-ray) radiation from the polar gaps can be much greater than in theabsence of positronium formation. We discuss this enhancement for (a)Arons-type models, in which particles flow freely from the surface, and findany enhancement to be modest, and (b) Ruderman-Sutherland-type models, in whichparticles are tightly bound to the surface, and find that the enhancement canbe substantial. In the latter case we argue for a self-consistent,time-independent model in which partial screening of $E_\parallel$ maintains itclose to the threshold value for field ionization of the bound pairs, and inwhich a reverse flux of accelerated particles maintains the polar cap at atemperature such that thermionic emission supplies the particles needed forthis screening. This model applies only in a restricted range of periods,$P_2<P<P_1$, and it implies an energy flux in high-energy particles that cancorrespond to a substantial fraction of the spin-down power of the pulsar.  Nonthermal, high-frequency radiation has been observed from six radio pulsarsComment: TEX file, 47 pages. Accepted by Australian J. Phy

Pulsars with Strong Magnetic Fields: Polar Gaps, Bound Pair Creation and Nonthermal Luminosities