DifferentialManipulatordifferential kinematics kinematics is the linear mapping between the generalized velocities of the robot and the corresponding task velocities. This mapping is characterized by a matrix called analytic Jacobian, which depends on the robot configuration. When considering the end-effector task for a manipulator, the geometric Jacobian describes the mapping between the joint velocities and the end-effector twist (linear velocity and angular velocity). The Jacobian matrix is the most important tool to analyze the instantaneous mobility of a robot and is used to solve the inverse differential kinematics problem. In this context, the kinematic singularities of the Jacobian (where mobility in task space is partially lost) are classified and studied for typical robot manipulator structures. In the presence of kinematic redundancy, the multiplicity of inverse solutions can be exploited by performing additional configuration space motions computed via optimization-based or task augmentation methods (including task prioritization), both at the velocity and acceleration levels. Schemes for closed-loop inverse kinematics are also presented, based on the Jacobian inverse, pseudoinverse, or transpose. In this context, different orientation errors are defined and their behavior is analyzed. Finally, it is shown that the Jacobian transpose also characterizes the statics of a manipulator, i.e., the mapping between the wrench (force and moment) applied to the end-effector and the resulting torques at the joints. The kineto-statics duality is highlighted, with application to twist and wrench transformations such as those needed for processing the data of a wrist force/torque sensor. This duality is also the basis for the definition of velocity and force manipulability ellipsoids, which conclude the chapter.