Approximate entropy (ApEn) is a recently formulated family of parameters and statistics quantifying regularity (orderliness) in serial data, with developments within theoretical mathematics as well as numerous applications to multiple biological contexts. We discuss the motivation for ApEn development, from the study of inappropriate application of dynamical systems (complexity) algorithms to general time-series settings. ApEn is scale invariant and model independent, evaluates both dominant and subordinant patterns in data, and discriminates series for which clear feature recognition is difficult. ApEn is applicable to systems with at least 50 data points and to broad classes of models: it can be applied to discriminate both general classes of correlated stochastic processes, as well as noisy deterministic systems. Moreover, ApEn is complementary to spectral and autocorrelation analyses, providing effective discriminatory capability in instances in which the aforementioned measures exhibit minimal distinctions. Representative ApEn applications to human aging studies, based on both heart rate and endocrinologic (hormonal secretory) time series, are featured. Heart rate (HR) studies include gender- and age-related changes in HR dynamics in older subjects, and analyses of "near-SIDS" infants. Endocrinologic applications establish clear quantitative changes in joint LH-testosterone secretory dynamics in older versus younger men (a "partial male menopause"), via cross-ApEn, a related two-variable asynchrony formulation; a disruption in LH-FSH-NPT (penile tumescence) synchrony in older subjects; and changes in LH-FSH secretory dynamics across menopause. The capability of ApEn to assess relatively subtle disruptions, typically found earlier in the history of a subject than mean and variance changes, holds the potential for enhanced preventative and earlier interventionist strategies.