To restore hearing sensation, cochlear implants deliver electrical pulses to the auditory nerve by relying on sophisticated signal processing algorithms that convert acoustic inputs to electrical stimuli. Although individuals fitted with cochlear implants perform well in quiet, in the presence of background noise, the speech intelligibility of cochlear implant listeners is more susceptible to background noise than that of normal hearing listeners. Traditionally, to increase performance in noise, single-microphone noise reduction strategies have been used. More recently, a number of approaches have suggested that speech intelligibility in noise can be improved further by making use of two or more microphones, instead. Processing strategies based on multiple microphones can better exploit the spatial diversity of speech and noise because such strategies rely mostly on spatial information about the relative position of competing sound sources. In this article, we identify and elucidate the most significant theoretical aspects that underpin single- and multi-microphone noise reduction strategies for cochlear implants. More analytically, we focus on strategies of both types that have been shown to be promising for use in current-generation implant devices. We present data from past and more recent studies, and furthermore we outline the direction that future research in the area of noise reduction for cochlear implants could follow.