It is a bit unfortunate that no single place for research-level maths videos yet exists (when one can even find a list of Paris places bearing the name of a mathematician). Here are some URLs that I could find:
[Edit: there's a very nice list of videos on Pinterest which focuses on courses and colloquium-style talks, some being taken from the pages below, but that list doesn't include most of the dozens of more specialized talks mentionned on those pages.]
MSRI (tons of workshops…)
IAS (includes Minerva Lectures…)
Newton Institute (tons of workshops…)
Institut Henri Poincaré (includes workshops and recent Bourbaki seminars…)
IHÉS (includes recent courses…)
PIMS (under the name mathtube.org)
ICTS (lots of courses…)
University of Oregon (includes Moursund Lectures).
Hebrew university of Jerusalem (includes Landau Lectures)
HIM (lots of seminars)
ICM (from 1998 Berlin ICM onwards)
U of Arizona (mostly Arithmetic Geometry)
IMA (tons of workshops…)
Math-net.ru (lots of seminars…)
U of Washington (includes Milliman Lectures…)
Michigan State U (includes Phillips Lectures…)
Topology seminars filmed by Carmen Rovi (UK, Oberwolfach…)
BIRS (lots of workshops…)
Institut Fourier (several summer schools…)
Feel free to mention other relevant ones in the comment section (the post will be periodically updated accordingly, with most recent additions at the end of the list).
Obiously, a nice global interface with search by keywords or names would be much more appropriate (I just might set it up later, if not too much work, and not too expensive).