Single-electron tunneling through zero-dimensional structures formed by lateral confinement of a two-dimensional electron gas is analyzed for both the vertical (current flow perpendicular to the quantum disk) and planar (current flow in the plane of the disk) structure. The discrete energy spectrum of the quantum dot influences drastically both the linear and the nonlinear transport through the dot. It is shown that the properties of a double dot system can be well described on the basis of a quantum molecule. Finally, a new approach for directly grown single-electron transistors is presented.