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 redundant! I'll make sure that any such modification in the README.md raised as an issue or submitted as a pull request will be timely reviewed and added.

The following are some global research internship positions:

1. MITACS Gloablink, for research positions in Canada.
2. DAAD Wise Scholarship, for research positions in Germany
3. Research fellowship at ETH Zurich
4. Max Planck Institute for Software Systems
5. CRUISE Program by Georgia Tech University
6. IPAM UCLA, RIPS Program
7. Summer at EPFL
8. Robotics Institute for Summer Scholars, CMU
9. SPIKE @ UNIST, Summer Program for Internship and Korean Experience
10. NTHU Summer Internship Program
11. Santa Fe Institute Summer Research Experience
12. UROP International, RWTH Aachen
13. University of Tokyo, Agmen Scholarship Japan (only Bio Tech and Bio-Chem Programs)
14. MTBI Summer Program, Arizona State University
15. Summer Undergraduate Research fellowship program, Rockfeller Institute
16. CERN Summer Student Program
17. CERN OpenLab for Computer Science Undergraduates
18. ISternship Summer Student Program, IST Austria
19. CalTech SURF Program
20. TIGP-IIP, Academia Sincia, Taiwan
21. Agmen Scholarship, University of Tokyo
22. IVISP, HKUST (only for senior year undergraduates + postgraduates)
23. Space Astronomy Summer Program, Space Telescope Science Institute (STScI)
24. UNSW Civil and Environmental Engineering Research Internship
25. OIST Research Internship Program, Okinawa Institute of Science and Technology
26. SURP, CUHK

The following are some research internships for Indian students:

1. S.N. Bose Scholarship, sponsored by IUSSTF
2. Shastri Research Student Fellowship by Shastri Indo-Canadian Institute
3. Viterbi India Program, sponsored by IUSSTF and USC Viterbi
4. Khorana Program for Scholars, sponsored by IUSSTF
5. NTU-India Connect, sponsored by NTU
6. Charpak Global Scholarship, funded by French Embassy in India
7. Indian Student Internship Program at NTHU
8. IBM Blue Mix Research Internship
9. IBM Extreme Blue (limited to IITs and NITs)
10. LIGO IndiGo at CalTech, funded by US NSF
11. SPARK, IIT Roorkee
12. IIITD Summer Intersnhip, IIIT, New Delhi
13. Summer Research Fellowship Programme 2018, IISc, IAS, INS
14. Quantitative Research Summer Internship, WorldQuant India
15. IITB Research Internship Award, IIT Bombay
16. IITK SURGE Program, IIT Kanpur