Matrix partition problems generalize graph colouring and homomorphism problems and occur frequently in the study of perfect graphs. It is difficult to decide, even for a small matrix M, whether the M-partition problem is polynomial time solvable or NP-complete (or possibly neither), and whether M-partitionable graphs can be characterized by a finite set of forbidden induced subgraphs (or perhaps some other first order condition). We discuss these problems for the class of chordal graphs.