We investigate surface deformation associated with currently active crustal magma bodies in Socorro, New Mexico, and Long Valley, California, USA. We invert available geodetic data from these locations to constrain the overall geometry and dynamics of the inferred deformation sources at depth. Our best-fitting model for the Socorro magma body is a sill with a depth of 19km, an effective diameter of 70km and a rate of increase in the excess magma pressure of 0.6kPayr-1. We show that the corresponding volumetric inflation rate is ~6×10-3km3yr-1, which is considerably less than previously suggested. The measured inflation rate of the Socorro magma body may result from a steady influx of magma from a deep source, or a volume increase associated with melting of the magma chamber roof (i.e. crustal anatexis). In the latter case, the most recent major injection of mantle-derived melts into the middle crust beneath Socorro may have occurred within the last several tens to several hundreds of years. The Synthetic Interferometric Aperture Radar (InSAR) data collected in the area of the Long Valley caldera, CA, between June 1996 and July 1998 reveal an intracaldera uplift with a maximum amplitude of ~11cm and a volume of 3.5×10-2km3. Modelling of the InSAR data suggests that the observed deformation might be due to either a sill-like magma body at a depth of ~12km or a pluton-like magma body at a depth of ~8km beneath the resurgent dome. Assuming that the caldera fill deforms as an isotropic linear elastic solid, a joint inversion of the InSAR data and two-colour laser geodimeter data (which provide independent constraints on horizontal displacements at the surface) suggests that the inferred magma chamber is a steeply dipping prolate spheroid with a depth of 7-9km and an aspect ratio in excess of 2:1. Our results highlight the need for large radar look angles and multiple look directions in future InSAR missions.