A simple electromagnetic medium is described by the constitutive relations D = ε E,B = μ H,J = σ E, where all the electromagnetic parameters, the permittivity ε, the permeability μ , and the conductivity σ are scalar constants. All other electromagnetic mediums are complex mediums. A sub-class of complex mediums are layered mediums which are piece-wise simple mediums. Each layer is a simple medium with scalar electromagnetic parameters. Availability of commercial software for electromagnetic simulation justifies reduction of time spent in teaching the mathematics of special functions in electromagnetic core course at the graduate level. However to make effective use of these packages, the ability to relate the dominant effect of each kind of a geometrical feature and the dominant effect of each kind of complexity in the medium properties could be important in interpreting the results and also to find out when the package did not give correct results. The course outline of a departmental core course in 'Electromagnetic Waves and Materials' at our university, which includes dispersive medium, anisotropic medium, chiral medium, periodic medium, superconducting medium and their applications in the areas of microwaves and optics will be presented. Extension of the system theory to the systems where the basic elements are complex mediums of various kinds is a challenging research area.