The isolation of a growing number of two-dimensional (2D) materials has inspired worldwide efforts to integrate distinct 2D materials into van der Waals (vdW) heterostructures. Given that any passivated, dangling-bond-free surface will interact with another through vdW forces, the vdW heterostructure concept can be extended to include the integration of 2D materials with non-2D materials that adhere primarily through non-covalent interactions. We present a succinct and critical survey of emerging mixed-dimensional (2D + nD, where n is 0, 1 or 3) heterostructure devices. By comparing and contrasting with all-2D vdW heterostructures as well as with competing conventional technologies, we highlight the challenges and opportunities for mixed-dimensional vdW heterostructures.