The “Overground” mentioned here , emmie, is a collection of routes, some formerly part of the National Rail network and some former London Underground routes which have now been taken under the control of TfL. They are:
Euston to Watford Junction
Richmond/Clapham Junction to Stratford
Barking to Gospel Oak
West Croydon/Crystal Palace/Clapham Junction/New Cross to Highbury & Islington
There is a nice map here:
http://www.tfl.gov.uk/assets/downloads/london-overground-network-map.pdf
As you can see, with the exception of the northern end of the Watford stretch, all are entirely within London Boroughs.