The cytoskeleton consists of networks of protein polymers which structurally and dynamically organize interiors of living cells. Microtubules exhibit a complex array of self-organization phenomena which are very sensitive to various laboratory conditions. In this paper we discuss the main features of microtubules focusing our attention on a selection of their physical properties, i.e., the questions of assembly dynamics and energy transfer along their protofilaments, the possible dipolar phases which we predict to exist, and, finally, the hypothesis of current flows associated with the electric field lines produced by cytoskeletal components.