Beginning with a straightforward formulation of electromagnetic 'cloaking' that reduces to a boundary value problem involving a single Maxwell first-order differential equation, explicit formulae for the relative permittivity–permeability dyadic and fields of spherical and circular cylindrical annular cloaks are derived in terms of general compressed radial coordinate functions. The general formulation is based on the requirements that the cloaking occurs for all possible incident fields and that the cloaks with frequency ω>0 have continuous tangential components of E and H fields across their outer surfaces, and zero normal components of D and B fields at their inner material surfaces. The tangential-field boundary conditions at the outer surface of the cloak ensure zero scattered fields, and the normal-field boundary conditions at the inner surface of the cloak are compatible with zero total fields inside the interior cavity of the cloak. For spherical cloaks, unlike cylindrical cloaks, these boundary conditions lead to all the tangential components of the E and H fields being continuously zero across their inner surfaces—cylindrical cloaks having delta functions in polarization densities at their inner surfaces. For H-wave incident fields, a nonmagnetic circular cylindrical annulus is found that has nonzero scattered fields but zero total fields within its interior cavity. For bodies with no interior free-space cavities, the formulation is used to derive nonscattering spherical and cylindrical 'concentrators' that magnify the incident fields near their centers. For static fields (ω=0), the boundary value formulation is appropriately modified to obtain a relative permeability dyadic that will cloak magnetostatic fields. Causality-energy conditions imply that, unlike magnetostatic cloaking, electrostatic cloaking as well as low-frequency cloaking for ω>0 is not realizable. It is also confirmed that perfect cloaking over any nonzero bandwidth violates causality-energy conditions and thus the cloaking of realistic time-dependent fields must be approximate.