Cochlear implant technology successfully restores hearing function to patients with sensory impairment. Although cochlear implant users generally hear well in quiet, they still find noisy conditions very challenging, hence the need to employ noise reduction algorithms in these systems to enhance the user experience. This paper reviews noise reduction algorithms in cochlear implants. Traditionally, such algorithms have been classified as either single- or multiple-channel, depending on the number of microphones they use. This review retains this general classification in looking at recent papers and extends it to reflect recent interest in machine learning techniques. The review concludes with consideration of promising future areas of research.