Optical waveguides are the key building block of optical fiber and photonic integrated circuit technology, which can benefit from active photonic manipulation to complement their passive guiding mechanisms. A number of emerging applications will require faster nanoscale waveguide circuits that produce stronger light-matter interactions and consume less power. Functionalities that rely on nonlinear optics are particularly attractive in terms of their femtosecond response times and terahertz bandwidth, but typically demand high powers or large footprints when using dielectrics alone. Plasmonic nanostructures have long promised to harness metals for truly nanoscale, energy-efficient nonlinear optics. Early excitement has settled into cautious optimism, and recent years have been marked by remarkable progress in enhancing a number of photonic circuit functions with nonlinear plasmonic waveguides across several application areas. This work presents an introductory review of nonlinear plasmonics in the context of guided-wave structures, followed by a comprehensive overview of related experiments and applications covering nonlinear light generation, all-optical signal processing, terahertz generation/detection, electro optics, quantum optics, and molecular sensing.