Critically ill patients that require kidney replacement therapy (KRT) are among the most ill and complex patients routinely encountered in the intensive care unit (ICU). Continuous KRT (CKRT) is used across many ICUs as the therapy of choice for hemodynamically unstable patients with kidney failure. Though existing trials have not shown superior survival or kidney recovery with CKRT relative to intermittent KRT, CKRT has largely become the standard of care in developed nations for the treatment of acute kidney injury (AKI) in patients with shock, acute brain injury, acute liver failure, and other forms of critical illness. As health care systems provide an ever-widening scope of organ-support therapies to increasingly complicated ICU patients, the use of CKRT is likely to expand. In this Core Curriculum, we review the physicochemical principles of CKRT, provide a comprehensive yet practical review of when and how to prescribe CKRT, and summarize seminal trials that serve as the foundations for our approaches to timing of initiation, dosing, vascular access, and anticoagulation for CKRT. We conclude by briefly highlighting a variety of essential, yet often underappreciated, components of the provision of high-value multidisciplinary care to patients receiving CKRT, including drug dosing, nutrition, physical rehabilitation, and CKRT quality assurance programs.