In nature we find many instances of complex behavior for example the dynamics of stock markets, power grids, internet networks, highway traffic, social networks, heartbeat dynamics, neural dynamics, dynamics of living organisms, etc. The study of these complex systems involves the use of tools of non-linear dynamics and non-equilibrium statistical physics. This dissertation is devoted to understanding two different sources of complex behavior – non-poissonian renewal events also called crucial events and infinite memory of fractional Brownian motion. They both generate 1/f noise frequency spectrum. Thus, we studied examples of both processes and also their joint action. We also tried to establish the role of crucial events in biological and physiological processes like biophoton emission during the germination of seeds, the dynamics of heartbeat and neural dynamics. Using a statistical method of analyzing the time series of bio signals we were able to quantify the complexity associated with the underlying dynamics of these processes. Finally, we adopted a model that unifies both crucial events and memory fluctuations to study the rhythmic behavior observed in heart rate variability of people during meditation. We were able to also quantify the level of stress reduction during meditation. The work presented in this dissertation …