Magnetoencephalography enables non-invasive detection of weak cerebral magnetic fields by utilizing super-conducting quantum interference devices (SQUIDs). Solving the MEG inverse problem requires reconstructing the locations and orientations of the underlying neuronal current sources based on the extracranial measurements. Most inverse problem solvers explicitly favor either spatially more focal or diffuse current source patterns. Naturally, in a situation where both focal and spatially extended sources are present, such reconstruction methods may yield inaccurate estimates. To address this problem, we propose a novel ComprEssive Neuromagnetic Tomography (CENT) method based on the assumption that the current sources are compressible. The compressibility is quantified by the joint sparsity of the source representation in the standard source space and in a transformed domain. The purpose of the transformation sparsity constraint is to incorporate local spatial structure adaptively by exploiting the natural redundancy of the source configurations in the transform domain. By combining these complementary constraints of standard and transformed domain sparsity we obtain source estimates, which are not only locally smooth and regular but also form globally separable clusters. In this work, we use the l(1)-norm as a measure of sparsity and convex optimization to yield compressive estimates in a computationally tractable manner. We study the Laplacian matrix (CENT(L)) and spherical wavelets (CENT(W)) as alternatives for the transformation in the compression constraint. In addition to the two prior constraints on the sources, we control the discrepancy between the modeled and measured data by restricting the power of residual error below a specified value. The results show that both CENT(L) and CENT(W) are capable of producing robust spatially regular source estimates with high computational efficiency. For simulated sources of focal, diffuse, or combined types, the CENT method shows better accuracy on estimating the source locations and spatial extents than the minimum l(1)-norm or minimum l(2)-norm constrained inverse solutions. Different transformations yield different benefits: By utilizing CENT with the Laplacian matrix it is possible to suppress physiologically atypical activations extending across two opposite banks of a deep sulcus. With the spherical wavelet transform CENT can improve the detection of two nearby yet not directly connected sources. As demonstrated by simulations, CENT is capable of reflecting the spatial extent for both focal and spatially extended current sources. The analysis of in vivo MEG data by CENT produces less physiologically inconsistent "clutter" current sources in somatosensory and auditory MEG measurements. Overall, the CENT method is demonstrated to be a promising tool for adaptive modeling of distributed neuronal currents associated with cognitive tasks.