We calculated the electronic structure and optical properties of Eu mono-doped ZnO systems with or without O vacancy. We also determined the relative energy of ferromagnetic and antiferromagnetic orders of Eu-double-doped ZnO systems. The double-doped systems possess high Curie temperature and achieve room temperature ferromagnetism. The magnetism in the Eu mono-doped system without O vacancy is caused by the -Eu3+-O2--Eu3+- bound magnetopolaron (BMP) model. The magnetism of Eu mono-doped ZnO systems with O vacancy is more stable than that without O vacancy, and such magnetism is attributed to the -Eu3+-VO++-Eu3+- BMP model. The absorption spectrum for mono-doped systems is red shifted, and this finding confirms that Eu-mono-doped ZnO is a candidate photocatalyst for various applications. Therefore, Eu-double-doped ZnO can be practically used as an unambiguous diluted magnetic semiconductor.