Electromagnetic boundaries, defined by the most general linear and local conditions, are considered in this paper. The conditions relate the normal components of the D and B vectors to the tangential components of the E and H vectors at each point of the boundary. Reflection of a plane wave from a planar boundary in an isotropic half space is analyzed and an analytic expression for the reflection dyadic is derived. It is shown that any plane wave can be decomposed in two components which do not interact in reflection. Properties of plane waves matched to the general boundary are given. Certain special cases, arising naturally from the general theory and labeled as E-boundary, H-boundary and EH-boundary conditions, are introduced as interesting novelties and some of their properties are studied. Other special cases with known results are considered in verifying the theory. A possible realization of the general boundary in terms of an interface of a bi-anisotropic medium is discussed in an Appendix.