Unmanned Aerial Vehicles (UAV) control has become a very important point of scientific study. The control design challenges of a UAV make it one of the most researched areas in modern control applications. This thesis specifically chooses the Quadrotor as the UAV platform. Considering the quadrotor has 4 rotors and 6 degrees of freedom, it is an underactuated system and is dynamically unstable that has to be stabilized by a suitable control algorithm in order to operate autonomously. This thesis focuses on the quaternion representation of the quadrotor system dynamics and develops an adaptive control for its trajectory tracking problem. The control design uses the certainty equivalence principle where adaptive tracking controls are designed separately for each of the translational and rotational subsystems. With this approach, the success of the outer loop translational control relies on the fast convergence of the inner loop rotational control in order to guarantee the system’s stability while achieving the tracking objective. For the translational subsystem in the outer loop, a modified geometric control technique is considered with an adaptive component for the estimation of the uncertain mass of the quadrotor. For the rotational subsystem in the inner loop a backstepping based control design is adopted due to its systematic design and intuitive approach. An adaptive component is further integrated with it to estimate the integrated components of the uncertain Moment of Inertia matrix and other constant parameters in the system dynamics to guarantee the stability of the inner loop system while achieving the tracking objective. Furthermore, a complete backstepping control design methodology is presented which overcomes the issues of certainty equivalence principle where the inner loop needs to execute significantly faster than the outer loop to stabilize the system.