Exponential integrators have long been used to solve initial value problems that have a linear part, and they have been established as being particularly advantageous for situations in which the linear part makes the system highly oscillatory or stiff. This seminar will explore the advantages of such methods in situations where the linear part makes the system dissipative or non-conservative. The most applicable types of systems are used to model conservative mechanics with added linear, possibly time-dependent, forcing and/or damping terms. Many such systems satisfy certain properties that precisely describe dissipation in energy, momentum, mass, 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. A damped/driven nonlinear Schrodinger equation serves as a useful example.