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I have polygon shapefile representing administrative districts. There are small spaces between polygons where the polygons should be touching. There are probably also overlapping polygons. How can I have the polygons snap together, filling the gaps or slivers and removing overlap (possibly in R, QGIS or SAGA)?

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You might be able to do this in R with the rmapshaper package, more specifically the ms_simplify function.

The best tool I've actually used for this is pprepair, which is a standalone C program, gettable from github:

https://github.com/tudelft3d/pprepair

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I'd use GRASS 7 for this personally, since its topologically aware. You can import with a snap distance set to remove small slivers, and then use the module v.clean in various ways to handle whatever other problems the software reports. As a last resort, you may have to hand edit a couple of points, but this should be unusual.

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