Digital coherent receivers have gained significant attention in the last decade. The reason for this is that coherent detection, along with digital signal processing (DSP) allows for substantial increase of the channel capacity by employing advanced detection techniques. In this paper, we first review coherent detection technique employed in the receiver as well as the required receiver structure. Subsequently, we describe the core part of the receiver — DSP algorithms — that are used for data processing. We cover all basic elements of a conventional coherent receiver DSP chain: deskew, orthonormaliation, chromatic dispersion compensation/nonlinear compensation, resampling and timing recovery, polarization demultiplexing and equalization, frequency and phase recovery, digital demodulation. We also describe novel subsystems of a digital coherent receiver: modulation format recognition and impairment mitigation via expectation maximization, which may gain popularity with increasing importance of autonomous networks.