Let L be an ample line bundle on a smooth projective variety X of dimension $n$. Demailly has introduced the Seshadri constant $\epsilon (\mathrm {L}, x)$ of L at $x$, which roughly speaking measures how positive L is at $x$. For example, if L is very ample, then $\epsilon (\mathrm {L}, x) \geq 1$ for all $x \in \mathrm {X}$. We study these invariants in the first non-trivial case, when X is a smooth surface. We prove (somewhat surprisingly) that in this case $\epsilon (\mathrm {L}, x) \geq 1$ for all except perhaps countably many $x \in \mathrm {X}$, and moreover if $\mathrm {L}\cdot \mathrm {L}> 1$ then the exceptional set is finite. On the other hand, simple examples due to Miranda show that $\epsilon (\mathrm {L}, x)$ can take on arbitrary small positive values at isolated points. The paper also contains some related examples and open problems.