Anomalous X-ray Pulsars (AXPs) belong to a class of neutron stars believed to harbor the strongest magnetic fields in the universe, as indicated by their energetic bursts and their rapid spindowns. We have developed a theoretical model that takes into account processes in the atmospheres and magnetospheres of ultramagnetic neutron stars, as well as the effects of their strong gravitational fields on the observable properties. Using this model, we have analyzed the X-ray spectra of a number of AXPs. We find that in all cases, the X-ray spectra are described very well with this emission model. The spectroscopically measured magnetic field strengths of these sources are in close agreement with the values inferred from their spindown properties and provide independent evidence for their magnetar nature. The analysis of spectral data using this physical model also sheds light on the long-term evolution of AXPs.