Startingfrom the basic theory of mechanisms, this lecture teaches the geometric and algebraicfundamentals of mechanisms and robotics based on necessary mathematical basissuch as vector algebra and matrix theory. The lecture teaches line geometry,including coordinates of points, vectors and lines, vector equation of lines, equationsof planes and plane coordinates, Plückerray coordinates and Plücker axis coordinates, duality of axis and ray coordinates,and line systems. Based on line geometry, the lecture introduces screw algebraand the Lie algebra, including screws, screw parameters, screw operation and screwalgebra, twists and Mozzi’s instantaneous screw axis, wrenches and Poinsot’s centralaxis theorem, analogy between instantaneous kinematics and statics, reciprocity,Lie algebras and the Lie bracket, Lie algebra representation, matrix commutator,and Jacobi identity.Thelecture further introduces finite displacement screws, operators, and Liegroups, including coordinate transformation and SE(3), the rotation operatorand the rotation group SO(3), the Euler-Rodrigues’ Formula and the exponential mapon SO(3), Rodrigues’ parameters, formulae and Cayley’s formula, rigid body displacementin quaternions and Lie groups, and the general displacement with dual quaternions.The lecture also introduces finite displacement-screw operators of SE(3) andLie group actions, including the finite displacement-screw operator and the adjointrepresentation of SE(3), finite displacement-screw matrix decomposition and thegeometrical interpretation, traces of the finite displacement-screw matrix and parametersof the finite displacement screw, finite displacement screw representations, derivativeof the finite displacement-screw matrix and the instantaneous screw of the Lie algebrase(3), and decomposition of Chasles’ motion represented by finite displacementscrews.Based on the teaching of these knowledges, the lecture teaches parallelmechanisms, followed by several famous modern mechanisms, including metamorphicmechanisms, origami mechanisms, and geometrical block-mechanisms. The aboveknowledges lay the foundation for the final lecture of the course and lead tothe application of mechanisms in robotics, including the deployable antenna,the metamaterial and the metamorphic walking robots.