We present results from a survey of X-ray emission properties of near main-sequence B stars, including several Be and β Cephei stars. The main conclusions of our survey are: 1) The X-rays are soft, probably because the shock velocity jumps are small since the terminal wind speeds are small. 2) A major fraction of the wind emission measure is hot, assuming wind theory estimates for the density distribution. A large fraction of the wind is not expected to be hot in current wind shock models. 3) A hard component is found to be present in τ Sco; possible causes are discussed. 4) For the Be stars, the X-rays emission is from a normal B-star wind that is coming from the poles as in the WCD model of Be stars. 5) None of the stars, including the β Cep stars, show noticeable variability in their X-rays. For the normal B stars we conclude from the lack of variability that the shocks are in the form of fragments in the wind instead of spherical shells. 6) Our observations suggest that all B stars are X-ray sources and that there is a basal amount of X-ray luminosity of about 10-8.5 L boi. The hot component in τ Sco and the high X-ray luminosity of B stars detected in the all-sky survey suggests that there is a source of X-ray emission in addition to wind shocks in some B stars.