This thesis will give a brief overview of some classical theorems of convexity, such as those of Tietze and Nakajima, which allow us to determine convexity based on local neighbourhoods of boundary points, and then proceed to introduce the concept of spindle convexity and r-convexity. I will then prove analogues of these important classical theorems for r-convexity, if possible. Lastly, I will go through two-point symmetrization for both convexity and r-convexity, and prove some new results for the r-convexity version of two-point symmetrization.