Here follows some links that I find very useful, starting with arxiv.org which is a depository of almost all maths and phys preprints since 1992. It's not easy to think of something that's not there.

arxiv.org

For almost anything that's not on arxiv.org, one can use google scholar, which is very useful in finding all papers that refer toa certain paper.

Google Scholar

Project Euclid includes papers that are not available on arxiv.org, including those that predate 1992.

Project Euclid

Not Even Wrong regularly posts links to conferences, books and papers on modern mathematical physics. Peter Woit provides an antidote to hype related to string theory as a unified theory of all fundamental interactions, but here we focus only on mathematical physics.

Not Even Wrong

Prof Takasaki's web site provides an introduction to integrable models.

Prof K Takasaki's integrable models web page

A number of research institutes post videos of lectures on modern mathematical physics, including

Online lectures, IAS, Princeton, NJ, USA

Online lectures, KITP, Santa Barbara, CA, USA

Online lectures, Perimeter Institute, Waterloo, Canada

Learn Mathematica

Euler Institute lecture series

In the following TASI lectures, I recommend M Aganagic's 2007 lectures on topological strings, as well as all lectures in 2015

TASI lecture videos

IHES, Paris

The Graduate Physics lectures are almost exclusively dedicated to modern developments in mathematical physics.

Graduate Physics lectures

A number of lecture series are posted on youtube.com

6D conformal field theory workshop

The 29th (2012) Jerusalem Winter School in Theoretical Physics

A number of maths and phys departments have channels on youtube.com where the post videos of their lecture series. The following are the ones that I know.

Stanford Institute for Theoretical Physics

Institute for Advanced Studies, Princeton

Department of Mathematics, Princeton

Department of Mathematics, Harvard

Department of Mathematics, U of Illinois

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