Internet data traffic capacity is rapidly reaching limits imposed by optical fiber nonlinear effects. Having almost exhausted available degrees of freedom to orthogonally multiplex data, the possibility is now being explored of using spatial modes of fibers to enhance data capacity. We demonstrate the viability of using the orbital angular momentum (OAM) of light to create orthogonal, spatially distinct streams of data-transmitting channels that are multiplexed in a single fiber. Over 1.1 kilometers of a specially designed optical fiber that minimizes mode coupling, we achieved 400-gigabits-per-second data transmission using four angular momentum modes at a single wavelength, and 1.6 terabits per second using two OAM modes over 10 wavelengths. These demonstrations suggest that OAM could provide an additional degree of freedom for data multiplexing in future fiber networks.