This dissertation is devoted to study of complex systems in human physiology particularly heartbeats and brain dynamics. We have studied the dynamics of heartbeats that has been a subject of investigation of two independent groups. The first group emphasized the multifractal nature of the heartbeat dynamics of healthy subjects, whereas the second group had established a close connection between healthy subjects and the occurrence of crucial events. We have analyzed the same set of data and established that in fact the heartbeats are characterized by the occurrence of crucial and Poisson events. An increase in the percentage of crucial events makes the multifractal spectrum broader, thereby bridging the results of the former group with the results of the latter group. The crucial events are characterized by a power index that signals the occurrence of 1/f noise for complex systems in the best physiological condition. These results led us to focus our analysis on the statistical properties of crucial events. We have adopted the same statistical analysis to study the statistical properties of the heartbeat dynamics of subjects practicing meditation. The heartbeats of people doing meditation are known to produce coherent fluctuations. In addition to this effect, we made the surprising discovery that meditation makes the heartbeat depart from the ideal condition of 1/f noise. We also discussed how to combine the wave-like nature of the dynamics of the brain with the existence of crucial events that are responsible for the 1/f noise. We showed that the anomalous scaling generated by the crucial events could be established by means of a direct analysis of raw data. The efficiency of the direct analysis procedure is made possible by the fact that periodicity and crucial events is the product of a spontaneous process of self-organization. We argue that the results of this study can be used to shed light into the nature of this process of self-organization.