Mikusinski's operational calculus is a method that uses ideas from abstract alge- bra, specifically commutative ring theory, to understand integral and derivative operators and thence to solve differential equations. This talk will show the algebraic and analytic background to the method, some historical information, how it applies to ordinary differen- tial equations of both integer-order and fractional types, and some recent work on applying it to coupled systems of fractional differential equations. The talk will be interdisciplinary within pure mathematics, touching on both analysis and algebra, and should be of interest to people from different fields.