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Based on first-principles theoretical calculations, we investigate the electronic structure of various defects in P-doped ZnO. We find that a P$_{\rm O}$ impurity occupying an O site is a deep acceptor while a P$_{\rm Zn}$ atom at a Zn site is the dominant donor, causing a compensation of acceptors. Under O-rich growth conditions, Zn vacancies (V$_{\rm Zn}$) are the main source of $p$-type conduction. Since V$_{\rm Zn}$ is mobile and strongly interacts with abundant P$_{\rm O}$ and P$_{\rm Zn}$ defects, resulting complexes, such as P$_{\rm Zn}$-2V$_{\rm Zn}$ and P$_{\rm O}$-V$_{\rm Zn}$, which behave as acceptors, are likely to be formed under non-equilibrium growth conditions, and are responsible for the $p$-type conduction. We also investigate the effect of the strong Coulomb repulsion for the Zn $d$ electrons on the electronic properties of various defects.