There has been much interest in blind (self-recovering) channel estimation and blind equalization where no training sequences are available or used. In multipoint networks, whenever a link from the server to one of the tributary stations is interrupted, it is clearly not feasible (or desirable) for the server to start sending a training sequence to re-establish a particular link. In digital communications over fading/multipath channels, a restart is required following a temporary path interruption due to severe fading. During on-line transmission impairment monitoring, the training sequences are obviously not supplied by the transmitter. Consequently, the importance of blind channel compensation research is also strongly supported by practical needs. We present a comprehensive summary of research development on single-user channel estimation and equalization, focusing on both training-based and blind approaches. Our emphasis is on linear time-invariant channels.