Two theorems of Sumihiro

We will discuss the proofs of two theorems of Sumihiro. The first one states that any normal variety $X$ with an action a connected linear algebraic group $G$ is covered by $G$-stable quasi-projective open subsets. If $G$ is a torus, then $X$ is covered by $G$-stable affine open subsets, and this is the second theorem.
The talk is based on the papers H. Sumihiro "Equivariant completion" and F. Knop, H. Kraft, D. Luna, T. Vust "Local properties of algebraic group actions".