Abstract: Exponential integrators have long been used to solve initial value problems that have a linear part, and they have been established as particularly advantageous for situations in which the linear part makes the system highly oscillatory or stiff. This seminar will exhibit the advantages of such methods in situations where the linear part makes the system dissipative or non-conservative. The types of equations that are most applicable are used to model mechanical systems with linear damping, which are otherwise conservative. Many such systems satisfy certain properties that precisely describe dissipation in energy, or momentum, or phase space area, or something else. The methods explored exactly preserve these properties; they can be constructed to arbitrarily high order; and they can be applied to many applications, including among others an oscillator with time-dependent damping, a forced-damped chaotic pendulum, a diffusionless Burgers-Fisher equation, and a damped nonlinear Schrodinger equation.