This paper surveys current state-of-the-art planning and control algorithms for self-driving urban vehicles, focusing on their application in urban settings. The objective is to provide a comprehensive overview of the techniques used in motion planning and control, highlighting their strengths and limitations. The paper discusses various approaches, differing in vehicle mobility models, environmental assumptions, and computational requirements. It presents a side-by-side comparison of these methods to assist in system-level design decisions. The decision-making hierarchy in autonomous vehicles is typically structured into four levels: route planning, behavioral decision making, motion planning, and vehicle control. Route planning involves selecting a path through the road network, while behavioral decision making determines appropriate driving actions based on the environment. Motion planning generates a feasible path or trajectory, and vehicle control ensures the vehicle follows the planned path. The paper reviews models used for motion planning and control, including the kinematic single-track model and models that account for inertial effects. The kinematic model assumes no-slip conditions and is suitable for low-speed planning, while models incorporating inertial effects are used for more accurate simulations. The paper also discusses various motion planning techniques, including variational methods, graph search methods, and incremental search techniques. For vehicle control, the paper examines methods for path and trajectory stabilization, including pure pursuit and feedback control strategies. It also discusses predictive control approaches and linear parameter varying controllers. The paper highlights the importance of robustness and stability in control systems, particularly in dynamic environments. The paper concludes by emphasizing the need for further research in motion planning and control for autonomous vehicles, particularly in handling complex urban environments. It also notes the importance of connected vehicle technology in improving safety and performance through information sharing and coordination between vehicles. The survey provides a detailed overview of the current state of the art in motion planning and control for self-driving cars, with a focus on systems operating at automation levels 3 and above.This paper surveys current state-of-the-art planning and control algorithms for self-driving urban vehicles, focusing on their application in urban settings. The objective is to provide a comprehensive overview of the techniques used in motion planning and control, highlighting their strengths and limitations. The paper discusses various approaches, differing in vehicle mobility models, environmental assumptions, and computational requirements. It presents a side-by-side comparison of these methods to assist in system-level design decisions. The decision-making hierarchy in autonomous vehicles is typically structured into four levels: route planning, behavioral decision making, motion planning, and vehicle control. Route planning involves selecting a path through the road network, while behavioral decision making determines appropriate driving actions based on the environment. Motion planning generates a feasible path or trajectory, and vehicle control ensures the vehicle follows the planned path. The paper reviews models used for motion planning and control, including the kinematic single-track model and models that account for inertial effects. The kinematic model assumes no-slip conditions and is suitable for low-speed planning, while models incorporating inertial effects are used for more accurate simulations. The paper also discusses various motion planning techniques, including variational methods, graph search methods, and incremental search techniques. For vehicle control, the paper examines methods for path and trajectory stabilization, including pure pursuit and feedback control strategies. It also discusses predictive control approaches and linear parameter varying controllers. The paper highlights the importance of robustness and stability in control systems, particularly in dynamic environments. The paper concludes by emphasizing the need for further research in motion planning and control for autonomous vehicles, particularly in handling complex urban environments. It also notes the importance of connected vehicle technology in improving safety and performance through information sharing and coordination between vehicles. The survey provides a detailed overview of the current state of the art in motion planning and control for self-driving cars, with a focus on systems operating at automation levels 3 and above.