The transport and chemical reactions of solutes are modelled as a cellular automation in which molecules of different species perform a random walk on a regular lattice and react according to a local probabilistic rule. The model describes advection and diffusion in a simple way, and as no restriction is placed on the number of particles at a lattice site, it is also able to describe a wide variety of chemical reactions. Assuming molecular chaos and a smooth density function, we obtain the standard reaction-transport equations in the continuum limit. Simulations on one- an two-dimensional lattices show that the discrete model can be used to approximate the solutions of the continuum equations. We discuss discrepancies which arise from correlations between molecules and how these disappear as the continuum limit is approached. Of particular interest are simulations displaying long-time behaviour which depends on long-wavelength statistical fluctuations not accounted for by the standard equations. The model is applied to the reactions a + b ⇌ c and a + b → c with homogenous and inhomogenous initial conditions as well as to systems subject to autocatalytic reactions and displaying spontaneous formation of spatial concentration patterns.