Cavity-enhanced direct frequency comb spectroscopy combines broad bandwidth, high spectral resolution, and ultrahigh detection sensitivity in one experimental platform based on an optical frequency comb efficiently coupled to a high-finesse cavity. The effective interaction length between light and matter is increased by the cavity, massively enhancing the sensitivity for measurement of optical losses. Individual comb components act as independent detection channels across a broad spectral window, providing rapid parallel processing. In this review we discuss the principles, the technology, and the first applications that demonstrate the enormous potential of this spectroscopic method. In particular, we describe various frequency comb sources, techniques for efficient coupling between comb and cavity, and detection schemes that utilize the technique's high-resolution, wide-bandwidth, and fast data-acquisition capabilities. We discuss a range of applications, including breath analysis for medical diagnosis, trace-impurity detection in specialty gases, and characterization of a supersonic jet of cold molecules.