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simpletest.py is the simple example given in the Basemap docstrings.

test.py exercises all the available map projections by plotting the etopo 
20 minute bathymetry/topography data using imshow.

contour_demo.py demonstrates the use of filled contours with map projections.

plotmap.py is the example on the matplotlib 'screenshots' page (included
in test.py) which shows the ETOPO topography as an image on a Lambert
Conformal projection (using imshow).

plotmap_pcolor.py plots the ETOPO topography using pcolor, on a Lambert 
Conformal and Orthographic projection.

nytolondon.py shows how to draw a great circle on a map (NY to London)

ireland.py draws maps of ireland with the crude, low and intermediate
resolution boundary datasets.

quiver_demo.py shows how to plot wind vectors on a map.

randompoints.py demonstrates the use of scatter to plot randomly distributed
points on the earth.

ortho_demo.py demonstrates the orthographic projection - you can interactively
enter the reference lat/lon (the viewpoint of the satellite).