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I am working with two polygon layers which overlap but are unrelated (one is congressional district boundaries, the other is MPO area boundaries).

I would like to generate a list of all the polygons in one layer which overlap with each polygon in the second layer. I.e. I want to generate a list of all the Congressional districts which are at least partially within each MPO area.

Which vector tools or plugins can I use to get there?

I'm using QGIS 1.8 on OSX 10.6.

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You can use the intersect tool in the Vector menu under Geoprocessing Tools. You select the two layers you are interested and an output location. This will create a new shapefile with the attributes of the overlapping features. In the attributes you should then have the MPO and Congressional District numbers.

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I would use spatial database (Spatialite for example) and SQL-querry for it. I suppose you know that QGIS is able to work with databases and run queries. Also there is a Spatialite-GUI that allow to add virtual shp-files to the database so you don't actually need to import your data into the database. Now if you need to get "ID" field from layer1 for objects that intersects with objects from layer2 use simple querry (assume that Geometry is the field where geometry is stored):

FROM layer1 
WHERE  (SELECT ST_Intersects(layer1.Geometry, layer2.Geometry)) 

Notice that no additional (unwanted) layers will be created here.

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