Agreed with chyantu except for the part,"10 days to land in another job after you have quit the job at one company"..Different people are giving different opinions on this, Please consult a lawyer before venturing out to new job..

When a person is in on H1B visa, he/she should be working for the US employer and getting paid according to the terms on the Labor Condition Application. There is no such thing as a grace period or bench time without pay. The employer has to pay the H1B employee whether or not the person has a "project." Anybody, could you post the link, where "10 days" grace period is available. As per terms, assumed that there is no grace period on H1, the last day of pay chex is the date of validity. Thanks

Chantyu is partially correct. If you are in H1B in a university, that falls in CAP exempt. To my understanding, as long as you move to another non profit research institute (only not for profit does not satisfy this criteria), you should be able to transfer your H1B. I am not a lawyer. You may want to contact an immigration lawyer too.

Thank you guys.... means it is risky to join Universities then....