X-ray emission from neutron stars provides insight to the nature of these objects. X-ray emission can arise from cooling radiation, which would help constrain neutron star equations of state, or from processes related to the magnetic braking of the pulsar, which can help determine pulsar magnetosphere properties. The youngest neutron stars (~103 years) have X-ray emission dominated by magnetospheric processes, while cooling emission has been detected from slightly older neutron stars (~104-105 years). I present X-ray observations of three young neutron stars, PSR 1509-58, PSR 0656+14, and PSR 1055-52. ROSAT Position Sensitive Proportional Counter observations of PSR 1509-58 in the supernova remnant MSH 15-52 confirm the presence of a compact nebula around the pulsar and reveal the magnetospheric nature of the pulsar emission. I then discuss ASCA observations covering the 0.5 to 10 keV X-ray range of the cooling neutron star candidates PSR 0656+14 and PSR 1055-52. Using combined ROSAT and ASCA spectra, I find that these two pulsars have evidence of X-ray emission from a heated polar cap. Finally, the results of a project to use ROSAT archival data to search for cooling neutron stars are discussed. Using the three cooling neutron star candidates (PSR 0656+14, PSR 1055-52, and Geminga) as standards, we develop criteria for selecting sources from a catalog of ROSAT PSPC point sources, and then study their temporal and spectral characteristics. FFTs on the sources produced no conclusive evidence of pulsations. Spectral fits of these sources are not able to identify sources as neutron stars, but we identify those sources that most closely resemble our standard stars spectrally. We have not detected any neutron stars in this search, and this is likely due to technical limitations. We identify the best candidates for further investigation with instruments in other wavelengths and with upcoming X-ray satellite missions.