Algebraic methods in connection with classical multidimensional geometry have proven to be very efficient in the computation of direct and inverse kinematics of mechanisms as well as the explanation of strange, pathological behaviour of mechanical systems. Generally one can say that every planar, spherical or spatial mechanism having revolute or prismatic joints can be described by systems of algebraic equations. In this talk we give an overview of the results achieved within the last years using algebraic geometric methods, geometric preprocessing and numerical analysis. We provide the mathematical and geometrical background, like Study's parametrization of the Euclidean motion group, the ideals belonging to mechanism constraints The methods are explained with different examples from mechanism analysis and synthesis.