In this talk, we will consider the mapping class group of a compact surface with n punctures, and its pure subgroup consisting of mapping classes that fix the punctures pointwise. We will describe two stability phenomena that the (co)homology of these families of groups (indexed by the number n of punctures) exhibit. When the surface has non-empty boundary, the `puncture' homological stability of the family of mapping class groups is closely related with the representation stability of the sequence of homology groups of the pure mapping class groups. If time permits, we will explain this relation and how it could be used to obtain new results.