A CCC limb and a new 3CCC parallel mechanism have been designed in this paper based on geometry analysis. Their mobility and geometrical constraints are discussed by using screw theory and geometrical equations separately. Following that both the inverse and forward kinematics of the 3CCC parallel mechanism are proposed, in which Dixon's resultant is used to get the forward solutions for the orientation and a eighth-order polynomial equation in one unknown is obtained, leading to the results for the position analysis, numerical examples confirm these theoretical results. A short comparison with the traditional Stewart platforms is presented in terms of kinematics, workspace and trajectory planning.