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The identification of the $\gamma$-ray source Geminga as a rotation-powered pulsar breaths new life into several of the oldest and most difficult problems in high-energy astrophysics. Among these are 1) the interpretation of the unidentified $\gamma$-ray sources in the Galactic plane, 2) the location and physics of $\gamma$-ray emission in rotation-powered pulsars, and 3) the study of magnetized neutron star atmospheres and cooling interiors via their thermal soft X-ray emission. Geminga is the first pulsar which is undetected in the radio. Apart from this unsurprising fact, Geminga is neither unique nor unusual. Rather, it is the prototype of a new class of pulsars whose X-ray and $\gamma$-ray properties are similar, and which have already afforded great progress in understanding the aforementioned issues. The discovery that Geminga and other middle-aged pulsars are capable of channeling a large fraction of their spin-down power into $\gamma$-rays bolsters the hypothesis that most of the unidentified $\gamma$-ray sources in the Galactic plane are pulsars, many of which could be radio-quiet. The high $\gamma$-ray efficiencies of Geminga and PSR 1055--52, together with their large soft X-ray modulation, supports the model in which $\gamma$-rays are generated in the outer magnetosphere of a dipole which is nearly orthogonal to the spin axis. The thermal X-ray emission is of special importance because it is the only source of information about the surface magnetic field geometry. All the soft X-ray pulsars for which high quality data exist have single-peaked profiles, which argues against a simple centered dipole. These same pulsars have an additional, tiny region of hotter thermal emission which can be interpreted as the open field line polar cap which is heated by particles flowing inward from the magnetosphere accelerator. Beyond the simplest blackbody interpretation of these data, magnetic effects on the opacity can modify the emergent spectra, and circumstellar resonant scattering can shape the pulse profiles. Recent observations and theories of these phenomena will be reviewed.
