  We generalize the notion of non-crossing partition on a disk to general
surfaces with boundary. For this, we consider a surface $\Sigma$ and introduce
the number $C_{\Sigma}(n)$ of non-crossing partitions of a set of $n$ points
laying on the boundary of $\Sigma$. Our proofs use bijective techniques arising
from map enumeration, joint with the symbolic method and singularity analysis
on generating functions. An outcome of our results is that the exponential
growth of $C_{\Sigma}(n)$ is the same as the one of the $n$-th Catalan number,
i.e., does not change when we move from the case where $\Sigma$ is a disk to
general surfaces with boundary.
