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The latest discovery of ferromagnetism in atomically thin films of semiconductors Cr2Ge2Te6 and CrI3 has unleashed numerous opportunities for fundamental physics of magnetism in two-dimensional (2D) limit and also for technological applications based on 2D magnetic materials. To exploit these 2D magnetic materials, however, the mechanisms that control their physical properties should be thoroughly understood. In this paper, we present a comprehensive theoretical study of the magnetic, electronic, optical and magneto-optical (MO) properties of multilayers (monolayer (ML), bilayer (BL) and trilayer) as well as bulk CrI3, based on the density functional theory with the generalized gradient approximation plus on-site Coulomb repulsion scheme. Interestingly, all the structures except the BL, are found to be single-spin ferromagnetic semiconductors. They all have a large out-of-plane magnetic anisotropy energy (MAE) of ∼0.5 meV/Cr, in contrast to the significantly thickness-dependent MAE in multilayers of Cr2Ge2Te6. These large MAEs suppress transverse spin fluctuations and thus stabilize long-range magnetic orders at finite temperatures down to the ML limit. They also exhibit strong MO effects with their Kerr and Faraday rotation angles being comparable to that of best-known bulk MO materials. The shape and position of the main features in the optical and MO spectra are found to be nearly thickness-independent although the magnitude of Kerr rotation angles increases monotonically with the film thickness. Magnetic transition temperatures estimated based on calculated exchange coupling parameters, calculated optical conductivity spectra, MO Kerr and Faraday rotation angles agree quite well with available experimental data. The calculated MAE as well as optical and MO properties are analyzed in terms of the calculated orbital-decomposed densities of states, band state symmetries and dipole selection rules. Our findings of large out-of-plane MAEs and strong MO effects in these single-spin ferromagnetic semiconducting CrI3 ultrathin films suggest that they will find valuable applications in semiconductor MO and spintronic nanodevices.