Covering matrices were introduced by Matteo Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In particular, he showed that PFA implies that certain covering matrices exhibit strong covering and reflection properties. In this series of talks, I will construct counterexamples to these covering and reflection properties and investigate their relationships with square principles. This will lead to an examination of a variety of square principles intermediate between square_kappa and square(kappa^+). In the first lecture, I will introduce the notion of a covering matrix and present results about the existence of certain types of kappa-covering matrices for kappa^+.