Most of real systems such as biological and social systems can be modeled by a complex network and elementary units of the system are displayed by nodes and edges represent interactions between them. In this thesis, we study The properties of complex networks and the disease spreading process. Disease propagation is one of the dynamical processes on the networks, which deals with the effect of network structure on the diseases spreading behavior and the dynamical phase transition. There are models to study disease spreading and epidemic such as SIS,SIR,... .Epidemic threshold is a phase transitionbetween the appearance of epidemic and the lake of it. We review the mean-field solution of this models brifly.mean-field solution give an proximate value of the epidemic threshold in the a complex network.The aim of this thesis is investigating the role of topology and properties of networks such as Erdos-Renyi network , scale free network with Barabasi-Albert model and the so-called achlioptas networks on the spreading of disease. We simulate the spreading process using the SIR model on these networks for power-law and random degree distributions.The results are compared for different network topologyes. It is shown that there are some similarities betwee achlioptas, scale free and random networks. Key words Complex networks, epidemic threshold, mean – feild theory, spreading process.