We calculate using the Boltzmann transport theory the density-dependent mobility of two-dimensional (2D) electrons in GaAs, SiGe, and AlAs quantum wells as well as of 2D holes in GaAs quantum wells. The goal is to precisely understand the recently reported breakthrough in achieving a record 2D mobility for electrons confined in a GaAs quantum well. Comparing our theory with the experimentally reported electron mobility in GaAs quantum wells, we conclude that the mobility is limited by unintentional background random charged impurities at an unprecedented low concentration of similar to 10(13) cm(-3). We find that this same low level of background disorder should lead to 2D GaAs hole and 2D AlAs electron mobilities of similar to 10(7) and similar to 4 x 10(7) cm(2)/V s, respectively, which are much higher theoretical limits than the currently achieved experimental values in these systems. We therefore conclude that the current GaAs hole and AlAs electron systems are much dirtier than the state-of-the-art 2D GaAs electron systems. We present theoretical results for 2D mobility as a function of density, effective mass, quantum-well width, and valley degeneracy, comparing with experimental data.