Recent observations of the X-ray and EUV emission of non-supergiant B stars are summarised. As compared with O stars, the X-rays of most of the near-main-sequence B stars are soft, and the stars show a departure from the L(x) = 10(-7) L(bot) relation. Using line driven wind models to provide an estimate of the density distribution, it is concluded that a major fraction of the wind emission measure is hot, whereas in shocked wind theory less than 10 percent of the wind emission measure should be hot. The X-ray observations suggest that all of the B stars are X-ray emitters with a basal X-ray luminosity of about 10 10(-8.5) L(bol). A hard component dominates the X-ray emission of tau Sco, and possible causes are discussed. For the Be stars, the X-ray emission is that which is expected from a normal B-star wind coming from the poles, as in the Wind Compressed Disk (WCD) model of Be stars. None of the stars, including the beta Cep stars, show noticeable variability in their X-rays. EUVE observations of epsilon CMa B2 II, find it to be the brightest object in the EUV sky at 500 to 700 A. It shows a Lyman continuum flux that is a factor of 30 higher than line blanketed model atmospheres. The continuum is seen on both sides of the He I 504 Angstrom edge, and the discrepancy with model atmospheres is even greater shortward of 504 Angstrom. The EUVE spectra show emission lines both from high stages of ionization (Fe IX to Fe XVI) and from low stages (HeII and O III). The He II Lyman alpha results from recombination following X-ray photoionization in the wind, and the O III resonance line is found to be present because of the Bowen fluorescence mechanism. Thus, there is and interesting coupling between the wind production by the EUV photospheric emission, the production of X-ray and line EUV emission by winds, and the production of fluorescence by recombination in the wind; all of these processes are now observable in B stars.