how to overlay shapefile and raster?

In R you can do

library(raster)
library(rgdal)
r <- raster('raster_filename')
p <- readOGR('shp_path', 'shp_file')
e <- extract(r, p, fun=mean)

e is a vector with the mean of the raster cell values for each polygon.

Following advice I got on the gdal-dev mailing list, I used StarSpan:

starspan --vector V --raster R1 R2 ... --stats mystats.csv avg mode

The results are saved to CSV format. At that time, that was already enough for me, but it should be possible to somehow forge a Shapefile from that info.

The following script allows you to do the task with GDAL: http://pcjericks.github.io/py-gdalogr-cookbook/raster_layers.html#calculate-zonal-statistics

# Calculates statistics (mean) on values of a raster within the zones of an polygon shapefile

import gdal, ogr, osr, numpy

def zonal_stats(input_value_raster, input_zone_polygon):

    # Open data
    raster = gdal.Open(input_value_raster)
    driver = ogr.GetDriverByName('ESRI Shapefile')
    shp = driver.Open(input_zone_polygon)
    lyr = shp.GetLayer()

    # get raster georeference info
    transform = raster.GetGeoTransform()
    xOrigin = transform[0]
    yOrigin = transform[3]
    pixelWidth = transform[1]
    pixelHeight = transform[5]

    # reproject geometry to same projection as raster
    sourceSR = lyr.GetSpatialRef()
    targetSR = osr.SpatialReference()
    targetSR.ImportFromWkt(raster.GetProjectionRef())
    coordTrans = osr.CoordinateTransformation(sourceSR,targetSR)
    feat = lyr.GetNextFeature()
    geom = feat.GetGeometryRef()
    geom.Transform(coordTrans)

    # Get extent of geometry
    ring = geom.GetGeometryRef(0)
    numpoints = ring.GetPointCount()
    pointsX = []; pointsY = []
    for p in range(numpoints):
            lon, lat, z = ring.GetPoint(p)
            pointsX.append(lon)
            pointsY.append(lat)
    xmin = min(pointsX)
    xmax = max(pointsX)
    ymin = min(pointsY)
    ymax = max(pointsY)

    # Specify offset and rows and columns to read
    xoff = int((xmin - xOrigin)/pixelWidth)
    yoff = int((yOrigin - ymax)/pixelWidth)
    xcount = int((xmax - xmin)/pixelWidth)+1
    ycount = int((ymax - ymin)/pixelWidth)+1

    # create memory target raster
    target_ds = gdal.GetDriverByName('MEM').Create('', xcount, ycount, gdal.GDT_Byte)
    target_ds.SetGeoTransform((
        xmin, pixelWidth, 0,
        ymax, 0, pixelHeight,
    ))

    # create for target raster the same projection as for the value raster
    raster_srs = osr.SpatialReference()
    raster_srs.ImportFromWkt(raster.GetProjectionRef())
    target_ds.SetProjection(raster_srs.ExportToWkt())

    # rasterize zone polygon to raster
    gdal.RasterizeLayer(target_ds, [1], lyr, burn_values=[1])

    # read raster as arrays
    banddataraster = raster.GetRasterBand(1)
    dataraster = banddataraster.ReadAsArray(xoff, yoff, xcount, ycount).astype(numpy.float)

    bandmask = target_ds.GetRasterBand(1)
    datamask = bandmask.ReadAsArray(0, 0, xcount, ycount).astype(numpy.float)

    # mask zone of raster
    zoneraster = numpy.ma.masked_array(dataraster,  numpy.logical_not(datamask))

    # calculate mean of zonal raster
    return numpy.mean(zoneraster)