A way of understanding the laws which govem the worId of elementary particIes has not been found yet. Present-day theoretical physicists have to be satisfied with compromises which, at the best, promise some success at the expense of generality and unity. U nder these circumstances a critical analysis of the basic concepts of modem quantum theory may be timely and usefuI. It is hoped that the value of such an analysis may be preserved even if, in the near future, new ways of understanding the basis of elementary particIe physics are discovered. In this monograph one specific aspect of this analysis is treated, namely the problems of geometry in the microworld. An out- line of geometrical measurements in the macroworld was given pre- viously. These measurements seem to be c1ear enough for at least a certain set of problems to be considered as a starting point for discussing the situation in the microworld. The concepts and methods which are useful in the macroworld may only indirectly be carried over into the microworld and they require a high degree of abstraction. In comprehending the physical content of dynamic variables which have geometric meaning, for example, the space-time partic1e coor- dinates x, y, z, t it is of ten necessary to have recourse to gedanken experiments which, although not feasible in practice, can nevertheless be compatible with the basic principles of geometry and quantum mechanics.