Publisher Summary This chapter discusses some circles of ideas and results that are believed to constitute progress on the theory of invariant subspaces. If a transitive algebra of linear continuous operators in a complex locally convex space contains a nonzero compact operator, then the commutant of this algebra consists of scalar operators. If a transitive algebra R contains a nonzero finite dimensional operator, then R is weakly dense in L(B). The examples of operators on Banach spaces with only trivial closed invariant subspaces are presented in this chapter. In Beauzamy and Read, one also constructs a norm on the polynomials, such that multiplication by x on the completion of this space is a transitive operator. As a part of the definition of the norm, one declares certain vectors small and certain operators small.