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There are plenty of fellowship programs that offer H1b visas. However, if you want to stay in this country, doing a 3 year fellowship after a 3 year residency on H1b will not help you. The 6 year limit will run out-you will have to go out for a year before applying for H1b visa again. Other options are to apply for a O-1 visa if you qualify, or to get the fellowship from a program that operates in an underserved area/ VA Hospital and apply for a NIW. You will get the EAD and possibly a conditional greencard before you finish the fellowship-and then if you work another 2 years in an MUA or HPSA, you will get your GC! If you are one of the lucky few that got a H-1b from an inner city/ VA program, you will have done 3 years of work that will be counted towards NIW and if you get a fellowship in a similar program, you can have a GC before you finish your fellowship! Hope this helps.