We classify all finite simple subgroups of the Cremona group Cr3(C).

The main goal of the article is to give the definition of algebraic stability that would permit us to consider stability, not only for algebraic vector bundles or torsion-free coherent sheaves, but for the abelian category of coherent sheaves or for whatever abelian category. We present an axiomatic description of the algebraic stability on an abelian category and prove some general results. Then the stability for coherent sheaves on a projective variety is constructed which generalizes Gieseker stability. Stabilities for graded modules and for quiver representations are also discussed. The constructions could be used for other abelian categories as well.