Abstract: The investigation of invariant subspaces is a natural first step in the attempt to understand the structure of operators. The powerful structure theorems that are known for finite-dimensional operators (the Jordan form) and normal operators (the spectral theorem) provide, in essence, decompositions into invariant subspaces of special kinds. However, for various important classes of Banach spaces and operators, the invariant subspace problem remains open. There exists a vast literature dedicated to this one of the most famous problems of functional analysis. The aim of the talk is to deal with some results in this area, chosen on an ad hoc basis in the setting of positive operators on Banach lattices, after giving a brief historical background including a few milestones. Some recent advances will be presented and a new result on the existence of an invariant ideal for a family of positive operators on locally convex solid Riesz spaces will be given.