is there any characterisation of all the holomorphic or meromorphic functions from the open unit disk to itself?

As an example of what I mean by characterisation, the holomorphic functions on the Riemann sphere are given by rational functions and the meromorphic functions on the torus are given by elliptic functions.

EDIT: as pointed out by Potato, the class of functions I am asking about is probably far too wide. So let me add the following two conditions:

1) the function is surjective

2) the function has a continuous extension to the unit circle

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    $\begingroup$ Almost certainly not. For starters, this class includes all biholomorphisms from $\mathbb D$ to sufficiently nice connected subsets of $\mathbb D$. $\endgroup$ – Potato Apr 18 '15 at 8:49

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