**Theorem:** 1. Every point of X has a G-stable open quasi-projective neighborhood.

2. If X is quasi-projective then it can be equivariantly embedded into a projective space.

The first proof uses the language of line bundles, the second field and valuation theory. In the last section, the Picard group of G is studied.}, address = {Basel-Boston}, author = {Knop, Friedrich and Kraft, Hanspeter and Luna, Domingo and Vust, Thierry}, booktitle = {Algebraische Transformationsgruppen und Invariantentheorie}, editor = {H. Kraft, P. Slodowy, T. Springer}, faupublication = {no}, pages = {63-76}, peerreviewed = {No}, publisher = {BirkhĂ¤user Verlag}, title = {{Local} properties of algebraic group actions}, volume = {13}, year = {1989} }