Bitcoin is a cryptographic peer-to-peer currency. It has been created in 2009 by Satoshi Nakamoto. In the Bitcoin system, there is no institution equivalent to a bank, for each user takes part in verifying the validity of a transaction. This system is said to be safe and to guarantee the ”pseudonymity” (anonymity through a pseudo, the Bitcoin address). In this document we will discuss the security of Bitcoin transactions and the anonymity of the users. We will study two attacks : the fast payment double spending attack and how using a lightweight Bitcoin client can endanger the privacy of its users. We will then experiment and analyze how these attacks work by setting up a simulation of Bitcoin transactions.