Optical frequency combs have the potential to revolutionize terabit communications(1). The generation of Kerr combs in nonlinear microresonators(2) is particularly promising(3), enabling line spacings of tens of gigahertz. However, such combs may exhibit strong phase noise(4-6), which has made high-speed data transmission impossible up to now. Here, we demonstrate that systematic adjustment of the pump conditions for low phase noise(4,7-9) enables coherent data transmission with advanced modulation formats that pose stringent requirements on the spectral purity of the comb. In a first experiment, we encode a data stream of 392 Gbit s(-1) on a Kerr comb using quadrature phase-shift keying and 16-state quadrature amplitude modulation. A second experiment demonstrates feedback stabilization of the comb and transmission of a 1.44 Tbit s(-1) data stream over up to 300 km. The results show that Kerr combs meet the highly demanding requirements of coherent communications and thus offer an attractive route towards chip-scale terabit-per-second transceivers.