We use a three-dimensional computational model to study the fluid transport and mixing due to the beating of an infinite array of cilia. In accord with recent experiments, we observe two distinct regions: a fluid transport region above the cilia and a fluid mixing region below the cilia tip. The metachronal wave due to phase differences between neighbouring cilia is known to enhance the fluid transport above the ciliary tip. In this work, we show that the metachronal wave also enhances the mixing rates in the sub-ciliary region, often simultaneously with the flow rate enhancement. Our results suggest that this simultaneous enhancement in transport and mixing is due to an enhancement in shear flow. As the flow above the cilia increases, the shear rate in the fluid increases and this shear enhances stretching, which is an essential ingredient for mixing. Estimates of the mixing time scale indicate that, compared to diffusion, the mixing due to the cilia beat may be significant and sometimes dominates chemical diffusion.