In this paper, we review the recent progress in the optical signal processing based on the nonlinearity of semiconductor optical amplifiers (SOAs). The four important optical signal processing functional blocks in optical switching are presented, i.e., optical wavelength conversion, optical regeneration, optical logic, and optical format conversion. We present a brief overview of optical wavelength conversion, and focus on various schemes to suppress the slow gain recovery of the SOA and improve the operating speed of the SOA-based optical switches. Optical regeneration including re-amplification, re-shaping and re-timing is also presented. Optical clock recovery that is essential for optical regeneration is reviewed. We also report the recent advances in optical logic and optical format conversion, respectively. After reviewing the four important optical signal processing functional blocks, the review concludes with the future research directions and photonic integration.