  Few layers Fe$_3$GeTe$_2$ is currently the only atomically thin ferromagnetic
metal, and thus has drawn huge attention in the field of two-dimensional (2D)
magnetism. In this paper, we perform a systematic first principle study on the
electronic structure, magnetic anisotropy energy (MAE), and magneto-optical
(MO) effects in monolayer (ML), bilayer (BL) and trilayer (TL) as well as bulk
Fe$_3$GeTe$_2$. All the considered structures of Fe$_3$GeTe$_2$ are predicted
to have large MAE of order $\sim$3.0 meV/f.u., being larger than reported 2D
ferromagnetic semiconductors Cr$_2$Ge$_2$Te$_6$ and CrI$_3$ and also being
comparable to that of FePt which has the largest MAE among the transition metal
alloys. This large MAE thus stabilizes the long range ferromagnetic order down
to atomically thin layers and also suggests promising applications of 2D
Fe$_3$GeTe$_2$ in high density data storage. Furthermore, the calculated
magneto-optical spectra show large magnetic circular dichroism, thus resulting
in large MO Kerr rotation and Faraday rotation angles. In visible frequency
range, Kerr rotation angles up to $\sim$1.0$^\circ$ for TL Fe$_3$GeTe$_2$ are
found. Such values are larger than famous MO transition metal alloy MnBi. Also,
large Faraday rotation angles are predicted for all considered Fe$_3$GeTe$_2$
structures. In particular, ML Fe$_3$GeTe$_2$ has a Faraday rotation angle of
-156$^\circ$/$\mu m$, which is three times larger than famous MO oxide
Bi$_3$Fe$_5$O$_{12}$. These important findings are analysed in terms of the
calculated orbital-decomposed density of states and dipole selection rule
derived from the group theory. Our findings thus suggest that few layers and
bulk Fe$_3$GeTe$_2$ are promising MO materials and could be widely applied to
nano MO devices in the future.
