Given the values observed on a plane parallel to a horizontal ground surface, solutions are obtained for the continuation of dynamic electromagnetic fields upward in air or downward into a conducting earth. The upward (away from secondary sources) continuation integrals for the real and imaginary parts of any electromagnetic field component with arbitrary frequency and in a medium with arbitrary electrical and magnetic constants are derived and simplified to the case where the conductivity is zero. However, for frequencies normally used in electromagnetic prospecting, the effect of displacement current is negligible and one does not need to use the rigorous formulas derived, because adequate accuracy can be obtained by using the simpler static field formulas for continuation in a nonconducting medium such as air.The central problem in electromagnetic continuation is one of extrapolating the observed field from one medium to another through a physical boundary, namely, the air-earth interface. From the magnetic field observed in air, one should be able to compute the same within the conducting earth. Conversely, from the electric field observed within the ground or on its surface, one should be in a position to calculate the same in air and also, of course, deeper into the ground. The continuity conditions for the vertical derivatives of the electromagnetic field components, which constitute the basis for continuing an electromagnetic field from one medium to another, are derived.Downward continuation formulas, suitable for practical use, are derived explicitly, through use of a Taylor expansion, for the vertical component of the magnetic field in air, this being the quantity which is commonly measured. Three-dimensional downward continuation formulations to depths of one and two units of grid spacing and two-dimensional continuation to a depth of one unit of grid spacing are derived under the assumption that the effect of displacement current can be neglected.