Methods featured are primarily conformal symplectic exponential time differencing methods, with a focus on families of methods, the construction of methods, and the features and advantages of methods, such as order, stability, and symmetry. Methods are applied to the problem of the damped harmonic oscillator. Construction of both exponential time differencing and integrating factor methods are discussed and contrasted. It is shown how to determine if a system of equations or a method is conformal symplectic with flow maps, how to determine if a method is symmetric by taking adjoints, and how to find the stability region of a method. Exponential time differencing Stormer-Verlet is derived and is shown as the example for how to find the order of a method using Taylor series. Runge-Kutta methods, partitioned exponential Runge-Kutta methods, and their associated tables are introduced, with versions of Euler's method serving as examples. Lobatto IIIA and IIIB methods also play a key role, as a new exponential trapezoid rule is derived. A new fourth order exponential time differencing method is derived using composition techniques. It is shown how to implement this method numerically, and thus it is analyzed for properties such as error, order of accuracy, and structure preservation.