I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..."

Then right after that he argued in favour of such a sentence. Could anyone tell me what did he mean exactly?. That is to say, in this case what is elliptic operator like, what is the heuristic idea which such a result relies on? ...and a little bit of more details about it.

As usual references will be appreciated.

ADD: Thanks for the comments below, but I think they do not answer the question of title : Why is RR an Index problem?. Up to this point, what I can see is that two numbers happened to be the same.