Continuum and soft robots can transform automation tasks requiring compliant interaction in constrained or unstructured environments, including healthcare, agriculture, marine, and space applications. However, their complex mechanics introduce significant challenges in modeling and control. Low-dimensional continuum mechanical models, such as rod theories, effectively capture the large deformations of slender bodies in contact-rich scenarios while balancing accuracy and computational efficiency. This paper presents a vertical survey of rod models for continuum and soft robots, spanning their mathematical foundations, robot modeling, and control applications. We review the main rod theories adopted in soft robotics and introduce a deformation-based classification of rod models for continuum and soft robots. Furthermore, we survey recent model-based and learning-based control strategies leveraging rod models, highlighting their role in manipulation and physical interaction tasks. Finally, we discuss advantages, limitations, research gaps, and emerging directions of rod-based approaches. This paper aims to serve as a reference for developing models and control strategies for continuum and soft robots.