A family X of sets of integers has the Ramsey Property if for every coloring of all infinite subsets of the integers into two colors there is a set in X all of whose infinite subsets receive the same color. Mathias used an inaccessible cardinal to construct a model of ZF in which every set has the Ramsey Property; it is a well-known open question whether Mathias's inaccessible is necessary. In the first part of the talk, we will discuss various known ways of constructing definable sets without the Ramsey Property and some related open questions. Then we will explore strengthenings of the Ramsey Property and some new work on the large-cardinal strength of these strengthenings. If time allows, we will prove a strong form of Ramsey’s Theorem due to Hrusak and examine more open questions.