I am making this list as an attempt to fill the void of any such open source list. Your contribution is vital to keep the list up-to-date for future students to use. Hence it's a humble request to each user to contribute any new opportunity you discover and update the old ones. Without updation, the web-links will break, and the list will become futile! I'll make sure that any such modification in the README.md raised as an issue or submitted as a pull request be timely reviewed and added.

The following are some global research internship positions:

CANADA

• MITACS Globalink, for research positions in Canada.

US

• NASA - CalTech Summer Undergraduate Research Fellowship (SURF), for research positions at Jet Propulsion Laboratory, California Institute of Technology.
• Santa Fe Institute Summer Research Experience
• Robotics Institute for Summer Scholars, CMU
• Data Science for Social Good Fellowship, CMU
• QRLSSP Summer Program, Arizona State University: Only for US Citizens / Permanent Residents
• Summer Undergraduate Research fellowship program, Rockfeller Institute
• CalTech SURF Program
• Space Astronomy Summer Program, Space Telescope Science Institute (STScI)
• The Smithsonian Institution Fellowship Program (SIFP)
• Data Visualization Programming Summer Student Internship in New York City, Flatiron Institute.
• Summer Research Program, Princeton University
• International Student Research Internship Program - McKelvey School of Engineering
• Luddy School of Informatics, Computing, and Engineering - Global Talent Attraction Program (GTAP)
• BioChemCoRe (Biology and Chemistry Computational Research), Amaro Lab, UC San Diego
• Global Research Experience in Advanced Technologies (GREAT) Program, UC Davis
• ICT Summer Research Program
• UC Berkeley Amgen Scholars program
• Summer Undergraduate Program in Engineering Research at Berkeley (SUPERB) (for Information Technology students
• SUMMER RESEARCH OPPORTUNITIES PROGRAM(SROP), Purdue University
• Summer Undergraduate Research Fellowship (SURF), Stanford University

SWITZERLAND

• Research fellowship at ETH Zurich
• CERN Summer Student Program
• CERN OpenLab for Computer Science Undergraduates
• E3, EPFL (Switzerland)
• Summer at EPFL

GERMANY

• Max Planck Institute for Software Systems
• Warwick Statistics Internship Scheme, University of Warwick
• DAAD WISE Scholarship
• HZDR Dresden, Summer student program (Germany)
• MaxSIP (Max Planck Institute Summer Internship)

UK

• DSSGx UK Summer Fellowship, University of Warwick
• Visual Geometry Group

INDIA

• ICTS LONG TERM VISITING STUDENTS PROGRAM for both International and Indian Students. Aims to provide opportunities to students of science, mathematics and engineering to spend a longer period of time (1 or 2 semesters).
• ICTS SN BHATT Memorial Excellence Fellowship for Undergraduate students of science, mathematics and engineering, who are in their third, fourth or fifth year of the program are eligible to apply.

SOUTH KOREA

• SPIKE @ UNIST, Summer Program for Internship and Korean Experience
• SPIKE@UNIST (South Korea)
• GIST Global Internship Program (South Korea)

TAIWAN

• NTHU Summer Internship Program
• UROP International, RWTH Aachen
• TIGP-IIP, Academia Sincia, Taiwan
• ITRI Global Internship program
• TEEP@Asia (Taiwan)
• TEEP Internship, Taiwan

JAPAN

• University of Tokyo, Amgen Scholarship Japan (only Bio Tech and Bio-Chem Programs)
• OIST Research Internship Program, Okinawa Institute of Science and Technology
• NIMS (Japan)
• Hennge Global Internship Program, for remote internship in a Japan-based tech start-up.

AUSTRIA

• ISternship Summer Student Program, IST Austria (Note: Indian students not eligible)

HONG KONG

• IVISP, HKUST (only for senior year undergraduates + postgraduates)
• SURP, CUHK
• SURP (Hongkong)

AUSTRALIA

• UNSW Civil and Environmental Engineering Research Internship
• UNSW Research Internship (Australia)

ISRAEL

• Summer Intern at the Kupcinet Getz International Summer Science School of the Weizmann Institute of Science

SPAIN

• Summer Training Program,CNIO

FRANCE

• OECD Internship Programme (France)

SAUDI ARABIA

• KAUST (Saudi Arabia)

MULTIPLE COUNTRIES

• IPAM UCLA, RIPS Program
• Amgen Scholars Program
• UX Research Internship, Red Hat
• Allen Institute for AI, Research and Engineering Internships

The following are some research internships for Indian students:

1. S.N. Bose Scholarship sponsored by IUSSTF, it's been closed for almost two years now.
2. ICTS Summer research program, by ICTS-TIFR.
3. DAAD Wise Scholarship, for research positions in Germany
4. Shastri Research Student Fellowship by Shastri Indo-Canadian Institute
5. Viterbi India Program, sponsored by IUSSTF and USC Viterbi
6. Khorana Program for Scholars, sponsored by IUSSTF
7. NTU-India Connect, sponsored by NTU
8. Charpak Global Scholarship, funded by French Embassy in India
9. Indian Student Internship Program at NTHU
10. IBM Blue Mix Research Internship
11. IBM Extreme Blue (limited to IITs and NITs)
12. LIGO IndiGo at CalTech, funded by US NSF
13. IIITD Summer Internship, IIIT, New Delhi
14. Summer Research Fellowship Programme 2021, IISc, IAS, INS
15. Quantitative Research Summer Internship, WorldQuant India
16. Internship At CMI, Chennai Mathematical Institute
17. Shastri Student Internship Project, Eligible universities here
18. Cisco Summer internship
19. Hackerrank SDE intern
20. SSERD Research Intern for research work in topics like Space Settlement, Astrophysics, Space Mission Design etc.
21. IISER Kolkata Summer Student Research Programme, IISER Kolkata
22. IITM Summer Fellowship Programme, IIT Madras
23. IITD GIPEDI, IIT Delhi
24. IITB Research Internship Award, IIT Bombay
25. IITK SURGE Program, IIT Kanpur
26. SPARK, IIT Roorkee
27. SRIP, IIT Gandhinagar
28. IIT Ropar Summer Internship [Information], IIT Ropar
29. Bhaba Atomic Research Centre
30. ISI Kolkata Summer Internship