Electronics obtained through the bottom-up approach of molecular-level control of material composition and structure may lead to devices and fabrication strategies not possible with top-down methods. This review presents a brief summary of bottom-up and hybrid bottom-up/top-down strategies for nanoelectronics with an emphasis on memories based on the crossbar motif. First, we will discuss representative electromechanical and resistance-change memory devices based on carbon nanotube and core-shell nanowire structures, respectively. These device structures show robust switching, promising performance metrics and the potential for terabit-scale density. Second, we will review architectures being developed for circuit-level integration, hybrid crossbar/CMOS circuits and array-based systems, including experimental demonstrations of key concepts such lithography-independent, chemically coded stochastic demultipluxers. Finally, bottom-up fabrication approaches, including the opportunity for assembly of three-dimensional, vertically integrated multifunctional circuits, will be critically discussed.