Electromagnetic (EM) waves are oscillating electric and magnetic fields. In this chapter, we develop the mathematics to describe their propagation. We begin with Maxwell equations. We show how Maxwell equations constitute a linear system. We show how we can solve for the modes of the linear system. These turn out to be plane waves. We describe the evolution of these plane waves. We then consider how waves travel across interfaces (Shevgaonkar, Electromagnetic Waves. Tata McGraw Hill, New Delhi, 2006). We consider interfaces between two dielectrics and interface between a dielectric and a conductor. In the former case, we see how waves change course when crossing an interface, and we recover the famous laws of refraction. In the latter case, we show how conductor reflects EM waves. This reflection forms the basis of mirrors in optics (Hecht, Optics. Pearson, London, 2014), waveguides, and cavities in microwave engineering (Das and Das, Microwave Engineering. McGraw Hill, New Delhi, 2015; Pozar, Microwave Engineering. Wiley, Singapore, 2003) as we see in the subsequent chapters.