Optically detected magnetic resonance using nitrogen-vacancy (NV) colour centres in diamond is a leading modality for nanoscale magnetic field imaging(1-3), as it provides single electron spin sensitivity(4), three-dimensional resolution better than 1 nm (ref. 5) and applicability to a wide range of physical(6-13) and biological(14,15) samples under ambient conditions. To date, however, NV-diamond magnetic imaging has been performed using 'real-space' techniques, which are either limited by optical diffraction to similar to 250 nm resolution(16) or require slow, point-by-point scanning for nanoscale resolution, for example, using an atomic force microscope(17), magnetic tip(5), or super-resolution optical imaging(15,15). Here, we introduce an alternative technique of Fourier magnetic imaging using NV-diamond. In analogy with conventional magnetic resonance imaging (MRI), we employ pulsed magnetic field gradients to phase-encode spatial information on NV electronic spins in wavenumber or 'k-spacer'(20) followed by a fast Fourier transform to yield real-space images with nanoscale resolution, wide field of view and compressed sensing speed-up.