We present an overview on the recent progress achieved on the controllable growth of diverse GeSi alloy nanostructures by molecular beam epitaxy. Prevailing theories for controlled growth of Ge nanostructures on patterned as well as inclined Si surfaces are outlined firstly, followed by reviews on the preferential growth of Ge nanoislands on patterned Si substrates, Ge nanowires and high density nanoislands grown on inclined Si surfaces, and the readily tunable Ge nanostructures on Si nanopillars. Ge nanostructures with controlled geometries, spatial distributions and densities, including two-dimensional ordered nanoislands, three-dimensional ordered quantum dot crystals, ordered nanorings, coupled quantum dot molecules, ordered nanowires and nanopillar alloys, are discussed in detail. A single Ge quantum dot-photonic crystal microcavity coupled optical emission device demonstration fabricated by using the preferentially grown Ge nanoisland technique is also introduced. Finally, we summarize the current technology status with a look at the future development trends and application challenges for controllable growth of Ge nanostructures.