Factorization of almost periodic (AP) matrix functions arises naturally in a
variety of problems, both theoretical and applied, and for many of them the
matrix in question is 2-by-2 and triangular. Even in this setting the
factorability properties remain a mystery, in striking difference with both
the scalar almost periodic case and with purely periodic matrix case.  
We will give a survey of currently available existential results regarding the
factorization of general AP matrix functions, as well as available approaches to 
constructive  factorization of the specific matrices mentioned above.