This textbook details the architecture of a digital coherent optical system and describes its main digital signal processing (DSP) algorithms. The authors first show how the  combination of advanced modulation techniques, DSP and  coherent detection  has led to significant gains in capacity and ease of operation. The authors follow the path of the information from its generation in the transmitter, to propagation through the fiber and processing by the DSP algorithms in the receiver. The work summarizes academic results and presents them in a didactic way to students and practitioners working on the area of  optical communications. A full suite of classroom materials is included for easy integration into a curriculum, containing theoretic and simulation problems, and off-the-shelf Matlab/Octave functions.