The 6-joint serial manipulators are thoroughly investigated from the viewpoint of their inverse kinematic solutions. To do this systematically, the manipulators are classified according to their joint arrangements and two topological concepts are introduced as type number and conjugacy. The conditions of validity are stated and the valid manipulators are further classified into four groups according to their type numbers as type-3, type-4, type-5, and type-6. The concept of conjugacy leads to a reduction in the number of manipulators to be investigated. The minimal conditions for the existence of analytical solutions are determined. The ways of obtaining analytical solutions are described for all typical samples of joint arrangement and type number classifications. If analytical solutions do not exist, then it is described how to obtain semi-analytical solutions of first, second, or at most third order. A kth order semi-analytical solution is such that the number of equations to be solved numerically can be reduced to k. Simplifications and manipulations of the kinematic equations both for their analytical or semi-analytical solutions and for the topological analysis of manipulators have been achieved by means of the numerous properties of the exponential rotation matrices. (C) 2002 Elsevier Science Ltd. All rights reserved.