We use data from the Sydney-AAO Multi-Object Integral Field Spectrograph Galaxy Survey and the Galaxy And Mass Assembly (GAMA) survey to investigate the spatially resolved signatures of the environmental quenching of star formation in galaxies. Using dust-corrected measurements of the distribution of H... emission, we measure the radial profiles of star formation in a sample of 201 star-forming galaxies covering three orders of magnitude in stellar mass (M*; 10 super( 8.1)-10 super( 10.95) M...) and in fifth nearest neighbour local environment density (...; 10 super( -1.3)-10 super( 2.1) Mpc super( -2)). We show that star formation rate gradients in galaxies are steeper in dense (log sub( 10)(.../Mpc super( 2)) > 0.5) environments by 0.58 plus or minus 0.29dexr sub( e) super( -1) in galaxies with stellar masses in the range 10 super( 10)<M*/M...<10 super( 11) and that this steepening is accompanied by a reduction in the integrated star formation rate. However, for any given stellar mass or environment density, the star formation morphology of galaxies shows large scatter. We also measure the degree to which the star formation is centrally concentrated using the unitless scale-radius ratio (r sub( 50,H...)/r sub( 50,cont)), which compares the extent of ongoing star formation to previous star formation. With this metric, we find that the fraction of galaxies with centrally concentrated star formation increases with environment density, from ~5 plus or minus 4 per cent in low-density environments (log sub( 10)(.../Mpc super( 2)) < 0.0) to 30 plus or minus 15 per cent in the highest density environments (log sub( 10)(.../Mpc super( 2)) > 1.0). These lines of evidence strongly suggest that with increasing local environment density, the star formation in galaxies is suppressed, and that this starts in their outskirts such that quenching occurs in an outside-in fashion in dense environments and is not instantaneous. (ProQuest: ... denotes formulae/symbols omitted.)