One of the most appealing qualities of additive manufacturing (AM) is the ability to produce complex geometries faster than most traditional methods. The trade-off for this advantage is that AM parts are extremely vulnerable to residual stresses (RSs), which may lead to geometrical distortions and quality inspection failures. Additionally, tensile RSs negatively impact the fatigue life and other mechanical performance characteristics of the parts in service. Therefore, in order for AM to cross the borders of prototyping toward a viable manufacturing process, the major challenge of RS development must be addressed. Different AM technologies contain many unique features and parameters, which influence the temperature gradients in the part and lead to development of RSs. The stresses formed in AM parts are typically observed to be compressive in the center of the part and tensile on the top layers. To mitigate these stresses, process parameters must be optimized, which requires exhaustive and costly experimentations. Alternative to experiments, holistic computational frameworks which can capture much of the physics while balancing computational costs are introduced for rapid and inexpensive investigation into development and prevention of RSs in AM. In this review, the focus is on metal additive manufacturing, referred to simply as “AM”, and, after a brief introduction to various AM technologies and thermoelastic mechanics, prior works on sources of RSs in AM are discussed. Furthermore, the state-of-the-art knowledge on RS measurement techniques, the influence of AM process parameters, current modeling approaches, and distortion prevention approaches are reported.