The first part of the course will give a brief an introduction to random walks on groups in the context of geometric group theory. We will investigate what feature of a group influences the behavior of random walks and what can be learned about the group from random walks. We will study the probability of return after n steps in the context of symmetric random walks which is captured by the convolution powers of the driving probability measure. In the second part of this course we will discuss the property of Rapid Decay and how it is used to obtain estimates of this important quantity, and we will see how this property is used in other fields. All those notions will be tested on classical examples and classes of groups.