Ongoing seminar series (with online attendance and/or recordings): 

BIMSA Integrable Systems Group:
https://sites.google.com/view/bimsa-iss
https://researchseminars.org/seminar/BIMSA-ISS

CRM Mathematical Physics Laboratory:
https://www.crmath.ca/en/mathematical-physics-laboratory/mathematical-physics-seminars/

Kyoto Soliton Laboratory:
https://www2.yukawa.kyoto-u.ac.jp/~kanehisa.takasaki/soliton-lab/index-e.html

Leeds Integrable Systems Group:
https://integrable-systems.leeds.ac.uk/seminar-activities/

Lisbon Integrability, Geometry, Asymptotics Seminar:
https://iga.research.ulisboa.pt/

Michigan: Integrable Systems and Random Matrix Theory:
https://sites.google.com/umich.edu/isrmt-seminar/home

Siberian Federal University : Mathematical Models and Integration Methods:
https://mmandim.blogspot.com/
https://researchseminars.org/seminar/mmandim

SISSA Mathematical Physics Seminar:
https://www.math.sissa.it/news/seminars/1
https://researchseminars.org/seminar/SISSAmath-ph

Yaroslav Centre for Integrable Systems:
https://cis.uniyar.ac.ru/workshops/intsys

Past seminar series (with recordings):

Geometry, Dynamics and Mechanics Seminar:
https://events.math.unipd.it/GDMSeminar/PreviousTalks

Integrability, Dualities and Deformations:
https://sites.google.com/view/intdualdef/archive

Junior Integrable Probability Seminar:
https://sites.google.com/view/junior-ips/talks

London Integrability Journal Club:
https://integrability-london.weebly.com/videos.html

Virtual Integrable Systems Seminar Series:
https://media.ed.ac.uk/playlist/dedicated/51612401/1_glh1xdfq/1_zl014ish

Send us your suggestions: You are very welcome to send us your suggestions for the above list. To do so, please send in your material by Email