MXene, a new kind of two-dimensional (2D) material, has a unique combination of excellent physical and chemical properties. Via computations on density functional theory and deformation potential theory, we investigated the electronic structure and predicted the carrier mobility of Ti2CO2 (a typical MXene) monolayers and nanoribbons. The Ti2CO2 monolayer is a semiconductor with a band gap of 0.91 eV, and the hole mobility in the monolayer reaches 10(4) orders of magnitude along both x and y directions, which is much higher than that of MoS2, while the electron mobility is about two orders of magnitude lower. The dramatic difference between the hole and electron mobilities also exists in nanoribbons. Moreover, our results suggest that width controlling and edge engineering would be effective in adjusting the carrier mobility of Ti2CO2 nanoribbons, and endow experimentally available Ti2CO2 with wide applications to field-effect transistors and photocatalysts.