I have got a number of polygons within a single larger polygon. Parts of the smaller polygons go out from under the larger one, and I want to delete the parts of the smaller polygons, that go outside the larger one. How can this be done?
You want to clip (QGIS -> Vector -> Geoprocessing Tool -> Clip) your smaller polygons with your larger polygon. This tutorial describes how to do this.
It also sounds like you might be able to use the Delete Part tool. For example, I have a countries.geojson. In it the United States polygon includes tons of tiny islands near Alaska. I want these gone and only the US mainland to remain. The Delete Part tool allows you to remove any of the marked polys by clicking on them (I already deleted Alaska itself). The tool will work on polygons both selected and unselected.