Collisions between Earth orbiting satellites and debris have been a topic of growing concern among satellite operators and governments for many years. At the heart of preventing collisions, which have been observed to have terrible consequences for the health of the space environment, is the timely identification of potential collisions and the accurate quantification of the probability of collision. This work will introduce novel methods for uncertainty propagation that take into account the collision geometry and adaptively respond to nonlinearity measures taken along the eigenvectors of the satellite state distributions in order to ensure proper conjunction algorithm selection and enhance computational efficiency. Local linear probability density function approximations are demonstrated to be appropriate for a wide class of collision scenarios and provide immense computational advantages over traditional conjunction analysis. Next, the effects of coordinate choice are explored. New formulations of the collision risk measure in spherical coordinates and orbital elements are derived and shown to provide increased accuracy over traditional conjunction analysis methods in Cartesian coordinates. These new formulations are made possible through the novel use of relative orbital elements, which are also instrumental in providing new insights into methods for identifying potential collisions and collision windows. Finally, new avenues for collision probability are explored through investigation of the steady state behavior of the collision risk measure. This analysis gives insight into the steady state distributions of orbiting objects and is used to develop upper bounds for the probability of collision between two satellites. These developments are then brought together in a single software tool called CRATER and compared against other contemporary approaches for conjunction analysis on a number of test cases.