This seminar is about the study of rare events for chaotic dynamical systems. We will address this issue by two approaches. One regards the existence of Extreme Value Laws (EVL) for stochastic processes obtained from dynamical systems, by evaluating a fixed random variable along the obits of the system. The other has to do with the phenomenon of recurrence to arbitrarily small (hence rare) events, which is commonly known as Hitting Time Statistics (HTS) and Return Time Statistics (RTS). We will show the connection between the two approaches both in the absence and presence of clustering. Clustering means that the occurrence of rare events have a tendency to appear concentrated in time. The strength of the clustering is quantified by the Extremal Index (EI), that takes values between 0 and 1. The stronger the clustering, the closer the EI is to 0. No clustering means that the EI equals 1. Using the connection between EVL and HTS/RTS we will interpret the existence of an EI less than 1 as due to the presence of underlying periodic phenomena.