We introduce the concept of biorthogonal bases of compactly supported matrix valued wavelets and examine the feasibility of their construction. Initially, we assume matrix filter banks assembled by FIR filters such that the perfect reconstruction property for matrix valued signals holds. Without admitting commutativity of any matrix multiplications and by imposing further conditions on the lowpass parts of the matrix filter banks, we construct compactly supported matrix valued wavelet functions that constitute biorthogonal Riesz bases for the matrix valued L2/(R, CN times/N/) signal space. To validate our design, we provide a class of biorthogonal compactly supported matrix valued wavelet bases. As a special case, a class of orthonormal compactly supported matrix valued wavelets is also constructed, which in turn leads to regular multiwavelet functions that possess symmetry, compact support and orthogonality properties.