Cetaceans have elicited the attention of researchers in recent decades due to their importance to the ecosystem and their economic values. They use sound for communication, echolocation and other social activities. Their sounds are highly non-stationary, transitory and range from short to long sounds. Passive acoustic monitoring (PAM) is a popular method used for monitoring cetaceans in their ecosystems. The volumes of data accumulated using PAM are usually big, so they are difficult to analyze using manual inspection. Therefore different techniques with mixed outcomes have been developed for the automatic detection and classification of signals of different cetacean species. So far, no single technique developed is perfect to detect and classify the vocalizations of over 82 known species due to variability in time-frequency, difference in the amplitude among species and within species' vocal repertoire, physical environment, among others. The accuracy of any detector or classifier depends on the technique adopted as well as the nature of the signal to be analyzed. In this article, we review the existing techniques for the automatic detection and classification of cetacean vocalizations. We categorize the surveyed techniques, while emphasizing the advantages and disadvantages of these techniques. The article suggests possible research directions that can improve existing detection and classification techniques. In addition, the article recommends other suitable techniques that can be used to analyze non-linear and non-stationary signals such as the cetaceans' signals. Several research have been dedicated to this topic, however, there is no review of these past results that gives a quick overview in the area of cetacean detection and classification. This review will help researchers and practitioners in the field to make insightful decisions based on their requirements.