This work provides an introduction to the theory of linear systems defined over rings. The theory is presented with a minimum of mathematical details, so that anyone having some knowledge of linear system theory should find the work easy to read. The presentation centers on four classes of systems that can be treated as linear systems over a ring. These are (i) discrete-time systems over a ring of scalars such as the integers; (ii) continuous-time systems containing time delays; (iii) large-scale discrete-time systems; (iv) time-varying discrete-time systems. The manuscript is available on the Web at http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19780019915_1978019915.pdf