The Fête de la Musique will be going on Saturday night, June 21.

Here's a link to find concerts by style and by location: Finding concerts (in French)

As a consequence, six metro lines will be running all night, though not stopping at all stations. The Noctilien's routes will change somewhat, and the trains and RATP will run throughout the night, stopping at all stations.
June 21 transport info and map

A special transport pass will be available for €2.50 which will be valid from 5 p.m. Saturday, June 21, until 7 a.m. Sunday, June 22.