A point $p$ in a plane continuum $X\subset {\mathbb{R}}^2$ is accessible if there exists an arc $A\subset {\mathbb{R}}^2$ such that $A\cap X=\{p\}$. I will describe our recent results about plane embeddings of continua and their accessible points. Specifically, I will discuss arc-like continua (the Nadler-Quinn problem), Knaster continua, and Ingram’s atriodic triod-like continuum. This is joint work with Andrea Ammerlaan and Ana Anušić.