QGIS: remove overlapping areas and fill gaps with the Processing toolbox

In polygon vector layers it is not unusual to find overlapped areas and gaps, especially if they were digitized without using the snapping or other editing tools that allow to avoid such errors. If this kind of errors are not too big they can all be easily removed using the QGIS Processing toolbox, specifically using one of the GRASS modules and "snapping" import option. In the screencast a snapping tolerance of 1 meter is used, this will quickly remove all the overlapped areas (and close the gaps) that have a width of less than 1m.