Magnetospheres of neutron stars are commonly modeled as either devoid of plasma in "vacuum'' models or filled with perfectly conducting plasma with negligible inertia in "force-free'' models. While numerically tractable, neither of these idealized limits can simultaneously account for both the plasma currents and the accelerating electric fields that are needed to explain the morphology and spectra of high-energy emission from pulsars. In this work we improve upon these models by considering the structure of magnetospheres filled with resistive plasma. We formulate Ohm's Law in the minimal velocity fluid frame and implement a time-dependent numerical code to construct a family of resistive solutions that smoothly bridges the gap between the vacuum and force-free magnetosphere solutions. We further apply our method to create a self-consistent model for the recently discovered intermittent pulsars that switch between two distinct states: an "on'', radio-loud state, and an "off'', radio-quiet state with lower spin-down luminosity. Essentially, we allow plasma to leak off open field lines in the absence of pair production in the "off'' state, reproducing observed differences in spin-down rates. Next, we examine models in which the high-energy emission from gamma-ray pulsars comes from reconnecting current sheets and layers near and beyond the light cylinder. The reconnected magnetic field provides a reservoir of energy that heats particles and can power high-energy synchrotron radiation. Emitting particles confined to the sheet naturally result in a strong caustic on the skymap and double peaked light curves for a broad range of observer angles. Interpulse bridge emission likely arises from interior to the light cylinder, along last open field lines that traverse the space between the polar caps and the current sheet. Finally, we apply our code to solve for the magnetospheric structure of merging neutron star binaries. We find that the scaling of electromagnetic luminosity with orbital angular velocity varies between the power 4 for nonspinning stars and the power 1.5 for rapidly spinning millisecond pulsars near contact. Our derived scalings and magnetospheres can be used to help understand electromagnetic signatures from merging neutron stars to be observed by Advanced LIGO.
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