In general cognitive systems are comprised of more than a single subprocess. The arrangement and linkages of these subprocesses are known as “mental architecture”. The simplest non-trivial systems or manner of carrying out multi-tasking on discrete items is found in two diametrically opposite models that have been classically used to describe the architecture of this processing. Serial systems allow only one subsystem at a time to operate. The antithesis of serial processing assumes that all subsystems operate simultaneously, or in parallel. Mathematical characterizations of these and other architectures have been developed. Models of serial vs. parallel systems, seemingly so distinct, are shockingly hard to tell apart with experimental methodology. Here we review the history of psychological work on the subject, both theoretical and experimental. We also describe three additional properties of systems that must be considered simultaneously (and have often been conflated with architecture): stopping rule, dependency, and capacity. We present formal mathematical descriptions that can allow us to examine in which cases the two will be indistinguishable, and summarize successful paradigms for differentiating the models in the context of various psychological tasks.