  The exact solution of the asymmetric exclusion problem and several of its
generalizations is obtained by a matrix product {\it ansatz}. Due to the
similarity of the master equation and the Schr\"odinger equation at imaginary
times the solution of these problems reduces to the diagonalization of a one
dimensional quantum Hamiltonian. We present initially the solution of the
problem when an arbitrary mixture of molecules, each of then having an
arbitrary size ($s=0,1,2, ...$) in units of lattice spacing, diffuses
asymmetrically on the lattice. The solution of the more general problem where
we have | the diffusion of particles belonging to $N$ distinct class of
particles ($c=1, ..., N$), with hierarchical order, and arbitrary sizes is also
solved. Our matrix product {\it ansatz} asserts that the amplitudes of an
arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed
by a product of matrices. The algebraic properties of the matrices defining the
{\it ansatz} depend on the particular associated Hamiltonian. The absence of
contradictions in the algebraic relations defining the algebra ensures the
exact integrability of the model. In the case of particles distributed in $N>2$
classes, the associativity of the above algebra implies the Yang-Baxter
relations of the exact integrable model.
