Binary black hole systems have three phases: inspiralling, merger and ringdown phases. This thesis is a review of methods developed to study the inspiralling phase analytically. Those methods include post Minkowskian (PM) and post Newtonian (PN) expansions of the metric, which are perturbative expansions in G and 1=c respectively. By applying these expansions the general solution with reasonable boundary conditions is derived, and its near and far zone limits are studied. This solution is observed to be a multipolar expansion, consisting of multipole moments that cannot be calculated directly. By using PN and PM expansions together, those multipole moments are calculated. At last, the metric is expanded to a 1 PN order and equations of motion for a binary black hole system in inspiralling phase is calculated to 1 PN order. Keywords: