In this talk, I will consider holomorphic self-maps of C^2 that fix the origin and are tangent to the identity (e.g., f(0) = 0 and df(0) = Id). An interesting topic to explore for such maps is how points near the origin move under iteration. Do they converge to the origin? If so, do they converge along a direction? I will discuss some results on the behavior of points near 0 under iteration by such a map. I will then focus on recent results of mine about what happens in a full neighborhood of the origin for a specific (degree two) map tangent to the identity. In addition, I will show how adding higher degree terms to this map will sometimes enable a domain of points to be attracted to the origin under iteration.