The problems of analysis and control of spatial behavior of dynamical systems associated with properties of invariance and attractivity of smooth geometric objects (goal sets) are discussed. A comparative study of the main concepts of geometric theory, coordinating and synergetic approaches is given and the problems of attractivity of invariant sets are considered in their connection with the partial stability analysis. Design procedures, static and dynamic control laws ensuring the desired properties of spatial dynamics are proposed, simplified local conditions of attractivity of the goal sets and partial stabilization of the system are represented.