Petri nets are powerful formalism for modeling a wide range of dynamic systems and system behaviors. This thesis surveys the basic concept and application of Petri nets. The structure of Petri nets, their marking and execution and several examples of Petri net modeling. In this thesis we research into the analysis of Petri nets. Also we give the structure of Reachability graphs of Petri nets and their advantages for analyzing the Petri nets. The reachability problem for Petri nets is the problem of finding if Mn 2 R(M0) for a given marking Mn in a net (N,M0).We present several different kinds of Petri nets, together with computer tools based on Mathematica. We give the Mathematica commands for Reachability problem and also we created Mathematica commands for Incidence matrix of Petri nets. We study the concept of Petri nets and applications of Petri nets.We especially focus on Biological applications on Petri nets and we work on modeling of Hashimoto.s Thyroiditis in Petri Nets.