Heartbeat fluctuations exhibit temporal structure with long-range correlations, fractal and nonlinear features, which break down with pathologic conditions, reflecting changes in the mechanism of neuroautonomic control. It has been hypothesized that these features change and also break down with advanced age, suggesting alterations in cardiac control with aging. Here we test this hypothesis. We analyze heartbeat RR intervals from two independent databases: (a) 19 young (avg. age 25.7 years) and 16 elderly healthy subjects (avg. age 73.8 years) during 2h rest from the Fantasia database; and (b) 29 healthy elderly subjects (avg. age 75.9 years) during 8h of sleep from the SHHS database, and the same subjects recorded 5 years later. We find: a significant difference in (1) the SD _RR and _RR of RR and their increments RR for the young and elderly Fantasia groups (P<10^-4) but no significant change with age for the elderly SHHS groups (P>0.5). In contrast, we find no significant difference in (2) the average heart rate RR (P>0.05), (3) the long-range correlation exponent _RR (P>0.15) (4) the fractal linear and nonlinear properties represented by the exponents ^sign (P>0.2) and ^mag (P>0.3) for the time series of the sign and magnitude of RR, and (5) the fractal dimension D(k) of RR. Our findings do not support the above-mentioned hypothesis. Our results show that the inherent temporal fractal and nonlinear organization of heartbeat fluctuations remains stable across time scales, indicating that the cascade of nonlinear feedback loops, underlying cardiac neuroautonomic regulation, remains intact with advanced age.