We live in a world that consist of different kinds of complex systems. Cosmological, economical, political, transportation, social and biological networks are among the most well known complex systems. One of the special features is related to understanding the evolutionary process of a new individual in a complex system. New individual may be a gene, pathogen, strategy, technique and behavioural pattern. If the new one could be able to take over the whole of a population, fixation would occur. Probability of this process is called fixation probability and the needed time for it is called fixation time . How individuals related to each other, that determined by the structure , affects on the fixation process. In this study, we focused on the effect of different structures on the fixation time. In the first step, we study the fixation time on simple structures, like Cycle, Complete and Star graphs. In this part we use the analytical and simulated approaches. We show that there are significant differences between fixation probability and fixation time, so that we cannot predict the behaviour of every one by understanding the other one. We studied some features of structures that affect on fixation time. Further we switch to complex networks, like Random, Small-World and Scale-Free networks. The behaviour of fixation time and the effect of structures are stud ied in this level, too. We figure out that the behaviour of fixation time is locally predictable by the knowledge of it from simple graphs. By this investigation we concentrated on fixation time in these systems by fixed fitness, however what we observe in real world is silently different. For that the individuals ’ fitness may change during the evolutionary process. By applying the game theory, we study the fixation time when the fitness changes for star graph specifically. The results of this thesis can be able to make a good landscape for understanding the behaviour of some quantities in evolutionary dynamics. This study might be a start point for the other studies in the evolutionary dynamics on the complex systems.