In this talk, I will discuss holomorphic self-maps in n complex dimensions that fix the origin and are tangent to the identity (i.e., f(0)=0 and df(0)=Id). I will give background in this area and discuss some of my new results. In particular, I will introduce a map in 2 complex dimensions that has 3 characteristic directions at the origin, but that does not have a domain of attraction along any of those directions. Instead, it exhibits other interesting dynamical behavior that I will discuss and supplement with pictures. I will then discuss joint work with F. Rong analyzing a family of maps tangent to the identity in 3 complex dimensions that have a characteristic direction whose directors have trivial real part and show that a domain of attraction does exist along that direction. Time permitting, I will show how small changes to both of these types of maps can affect the existence of a domain of attraction. Speaker(s): Sara Lapan (UC Riverside)