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I have downloaded OSM raster data from the openlayers plugin on QGIS, and put this on the map canvas using the web menu button. The projection is EPSG 3395 WGS world Mercator. However the projection for the vector data is EPSG 32633 WGS 84 UTM time zone 33N (Norway). Using the properties button on each layer, I cannot find the options to get the same CRS for each layer. The polygons in the vector layer seem out of place with the raster layer.

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    The OSM raster data is usually EPSG:3857, so try Set CRS for Layer (not reprojection or Save As). How much offset (in meters) do you have? Maybe the vector data is not WGS84 UTM, but ED50 UTM or NGO 1948 UTM. – AndreJ Jan 19 '18 at 6:09
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As AndreJ commented, first you have to make sure you have SET the correct CRS for each layer in the layer properties menu. This means that the coordinates of each layer are being read using the proper CRS. (changing this DOES NOT change the layer coordinates)

If this is done correctly, having the "on the fly" activated should display the layers together.

Not using the "on the fly" (which I recommend) means you will have to reproject a layer, using the tool "project" from the toolbox.

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