Given a divisor "D" on a compact Riemann surface "X", it is important to study the complex vector space of meromorphic functions on "X" with poles at most given by "D", called "H"0("X", "O"("D")) or the space of sections of the line bundle associated to "D". The degree of "D" says a lot about the dimension of this vector space. For example, if "D" has negative degree, then this vector space is zero (because a meromorphic function cannot have more zeros than poles). If "D" has positive degree, then the dimension of "H"0("X", "O"("mD")) grows linearly in "m" for "m" sufficiently large. The Riemann-Roch theorem is a more precise statement along these lines. On the other hand, the precise dimension of "H"0("X", "O"("D")) for divisors "D" of low degree is subtle, and not completely determined by the degree of "D". The distinctive features of a compact Riemann surface are reflected in these dimensions.