We report on β-Ga2O3 fin-shape transistors in a lateral geometry. The fin channel devices were fabricated on MOCVD-grown lightly Si-doped (010) β-Ga2O3 films. A hot phosphoric acid etch was used to remove dry etch damage and for the fabrication of sub-micron channels. Lateral fin-channel MESFETs and MISFETs with different fin widths were systematically studied. Electron conduction in the MISFET devices was found to be dominated by electron accumulation at the dielectric/semiconductor interface. In addition, the MISFETs showed higher drain-induced barrier lowering as compared to the MESFET devices. Comparison of the MISFETs with different fin widths suggested the importance of thin fin channels for better gate control and higher breakdown voltages.