We have overgrown GaAs (0 0 1) substrates, patterned with dense square arrays of round-shaped nanometer-sized holes, with GaAs and In(Ga)As. For GaAs overgrowth, the initial holes transform into larger well-defined multicornered holes. A subsequent deposition of InGaAs or InAs onto this template causes the formation of an ordered array of laterally closely spaced In(Ga)As quantum dots (QDs)-termed QD molecules. For GaAs buffer layers thicker than 18 monolayer, the QD molecules tend to align in [1 1 0] direction. On the other hand, if we overgrow the patterned hole array directly with InGaAs, we observe the formation of [(1) over bar 1 0]-aligned QD molecules. Overgrowth of such QD molecule arrays. with a Ga(Al)As spacer and, a second InGaAs QD layer results in the formation of about 1 million perfectly site-controlled InGaAs QDs. Furthermore, we investigate the photoluminescence property of a vertically and laterally aligned InAs QD array and simulate the strain energy density distribution generated by the buried QDs. (C) 2003 Elsevier B.V. All rights reserved.