This paper describes completely the invariant subspaces of the operator of multiplication by a finite Blaschke factor on the Banach space BMOA of analytic functions with bounded mean oscillation on the unit circle in the complex plane. As a simple application, we describe by very elementary means, the invariant subspaces of the co-analytic Toeplitz operator T_\overline{B} on H¹. In the simplest case when B(z) = z, the invariant subspaces of T_\overline{B} on H¹ were described by fairly deep arguments until the appearance of an elementary proof by two of the authors (Sahni & Singh). In recent times, the common invariant subspaces of the operators of multiplication by B² and B³, first in the case of z² and z³, and then for an arbitrary finite Blaschke B, have proved to be critical in the context of Nevanlinna-Pick type interpolation on H². Thus, keeping in mind the importance of invariant subspaces, we also offer a characterization of the common invariant subspaces of these operators on BMOA. Our proofs are that much more technical. Again, as an application, we obtain the common invariant subspaces of T_{\overline{B²}} and T_{\overline{B³}} on the Hardy space H¹.