Optical frequency combs have the potential to revolutionize terabit communications1. Generation of Kerr combs in nonlinear microresonators2 represents a particularly promising option3 enabling line spacings of tens of GHz. However, such combs may exhibit strong phase noise4-6, which has made high-speed data transmission impossible up to now. Here we demonstrate that systematic adjustment of pump conditions for low phase noise4,7-9 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 on a Kerr comb using quadrature phase shift keying (QPSK) and 16-state quadrature amplitude modulation (16QAM). A second experiment demonstrates feedback-stabilization of the comb and transmission of a 1.44 Tbit/s 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 solution towards chip-scale terabit/s transceivers.