Imagine that an insurance company collects premiums at a steady rate of per month. Let be the random amount that the insurance company has to pay out in month to cover customer claims. Let e the total pay-out in n months. Naturally the premiums must cover the average outlays, so . The company stays solvent as long as . Quantifying the chances of the rare event are then of obvious interest. The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter for example the event is the above. We also for example when we are concern on the number of random components of a system, the time over which a stochastic system is observed, or the temperature of a chemical reaction. The theory has applications in many different scientific fields, ranging from queuing theory to statistics and from finance to engineering. Poisson cluster processes are one of the most important justify; LINE-HEIGHT: normal; MARGIN: 9pt 0cm; unicode-bidi: embed; DIRECTION: ltr" this paper we derive scalar and sample path large deviation principles for Poisson cluster processes. Our results are potentially useful to study risk processes with Poisson cluster arrivals and light-tailed claims. Particularly, they may lead to determine the asymptotic behavior of the ruin probability, the most likely path to ruin and an efficient Monte Carlo algorithm to estimate the ruin probability. Results in this direction can be found in Stabile and Torrisi, where risk processes with non-stationary Hawkes arrivals are studied. This thesis is organized as follows. In chapter 1 , first we review describe the history of Large deviations and the Poisson cluster process and then briefly discuss some preliminaries on the theory of large deviations. Finally, some basic definitions and theorems are expressed. In chapter 2 , the theory of large deviation and some related theorems of large deviation in Poisson cluster processes and Hawkes processes are stated by expressing some examples. In chapter 3 , first we give some preliminaries on Poisson cluster processes, Hawkes processes and large deviations and then we state some theorems on large deviations in the Poisson cluster processes and Hawkes processes. In the last chapter, the results obtained in the previous chapter are illustrated by simulations.