One of the most ubiquitous objects in mathematical physics is the electromagnetic (field) tensor, not surprisingly given its relationship to Maxwell's equations, which are oftentimes known as the greatest equations ever discovered. I hope to talk about quantum field theories in a future talk, but it can be illuminating to study an example of a classical field theory like electromagnetism first. In this talk, I plan to say as much about the electromagnetic tensor as possible in a short time. Part of the talk will be physics - seeing it written out explicitly, using it to write Maxwell's equations in a coordinate-invariant way, using it to derive Maxwell's equations from the equations of motion, etc. And part of the talk will be math/mathematical physics - for example for this latter idea of the equations of motion, we can make the notion of the action function mathematically precise. A couple of short digressions into elementary de Rham cohomology and Hodge theory will be made to more rigorously rewrite Maxwell's equations using the electromagnetic tensor. Time permitting, we'll do an interesting problem or two at the end. There really aren't any prerequisites for this talk other than calculus and linear algebra, although knowing what a "tensor" is or "tensor notation" would be very helpful.