Line-intensity mapping (LIM) is an emerging approach to survey the Universe, using relatively low-aperture instruments to scan large portions of the sky and collect the total spectral-line emission from galaxies and the intergalactic medium. Mapping the intensity fluctuations of an array of lines offers a unique opportunity to probe redshifts well beyond the reach of other cosmological observations, access regimes that cannot be explored otherwise, and exploit the enormous potential of cross-correlations with other measurements. This promises to deepen our understanding of various questions related to galaxy formation and evolution, cosmology, and fundamental physics. Here we focus on lines ranging from microwave to optical frequencies, the emission of which is related to star formation in galaxies across cosmic history. Over the next decade, LIM will transition from a pathfinder era of first detections to an early-science era where data from more than a dozen missions will be harvested to yield new insights and discoveries. This review discusses the primary target lines for these missions, describes the different approaches to modeling their intensities and fluctuations, surveys the scientific prospects of their measurement, presents the formalism behind the statistical methods to analyze the data, and motivates the opportunities for synergy with other observables. Our goal is to provide a pedagogical introduction to the field for non-experts, as well as to serve as a comprehensive reference for specialists.