Lab-on-a-chip tools have played a pivotal role in advancing modern biology and medicine. A key goal in this field is to precisely transport single particles and cells to specific locations on a chip for quantitative analysis. To address this large and growing need, magnetophoretic circuits have been developed in the last decade to manipulate a large number of single bioparticles in a parallel and highly controlled manner. Inspired by electrical circuits, magnetophoretic circuits are composed of passive and active circuit elements to offer commensurate levels of control and automation for transporting individual bioparticles. These specifications make them unique compared to other technologies in addressing crucial bioanalytical applications and answering fundamental questions buried in highly heterogeneous cell populations. In this comprehensive review, we describe key theoretical considerations for manufacturing and simulating magnetophoretic circuits. We provide a detailed tutorial for operating magnetophoretic devices containing different circuit elements (e.g., conductors, diodes, capacitors, and transistors). Finally, we provide a critical comparison of the utility of these devices to other microchip-based platforms for cellular manipulation, and discuss how they may address unmet needs in single-cell biology and medicine.