The autonomous driving industry is experiencing unprecedented growth, driven by rapid advancements in technology and increasing demand for safer, more efficient transportation. At the heart of this revolution are two critical factors: lateral and longitudinal controls, which together enable vehicles to track complex environments with high accuracy and minimal errors. This paper provides a detailed overview of two of the field's most commonly used and stable control algorithms: proportional-integral-derivative (PID) and pure pursuit. These algorithms have proved useful in solving the issues of lateral (steering) and longitudinal (speed and distance) control in autonomous vehicles. This survey aims to provide researchers, engineers, and industry professionals with an in depth understanding of these fundamental control algorithms, their current applications, and their potential to shape the future of autonomous driving technology.