As Americans possessing a French visa, we must register with our local OFII and obtain the vignette OFII as soon as we come to France. Chances are you've already completed this process, but did you know that the combination of your visa and vignette OFII serves as a titre de séjour? This is an important idea for understanding the visa renewal process.
To be legal in France, you are required to have a titre de séjour, which is typically an identity card communicating information about you to relevant French authorities. As first year Americans, we don't receive this card and instead receive a use the combination of our visa and the vignette OFII to form an equivalent titre de séjour. I suppose the reason is to reduce administrative tasks surrounding the titre de séjour, but, let's be honest: the administrative tasks are probably more tedious for French civil servants, mais bon.
Prolonging your stay beyond your visa end-date is possible and simple if your reason for staying is of the same nature as your visa. A teaching assistant who came on a work visa, for example, and who plans on renewing a work contract with his or her school; a student with a student visa who wishes to continue his or her studies into a second year: these are reasons to stay that are of the same nature of the visas with which they came. In this case, to renew the titre de séjour, you need to go to your local préfecture or sous-préfecture (if your department is organized as such) and provide them with the documents they require.
If you would like to stay in France but change your reason for staying, this process is more complicated, more expensive, and more time-consuming--but it is possible. Personally, I did the teaching assistant program while beginning a master's degree at a French university in Paris, so, when I wanted to renew my titre de séjour, I was told that I couldn't. I couldn't because I had a work visa and my reason for staying was to continue my studies--i.e., the nature of my stay had changed. (An exception to this rule: students graduating from a French institution of higher education have the right to look for work after their studies.) The solution to this problem is to return to the US with your grades and an admission note and apply for a student visa at your local consulate. Then, when you return to France, you make an appointment with the OFII--and you're already familiar with the rest! You'll be good for another year.
A good piece of advise from anyone who has gone through the agony of French administrative tasks is to be aware of the requirements, to bring everything (as well as copies) plus anything else you might have. It just helps to be overly prepared.
