Abstract: Volume preserving mean curvature flow was introduced by Huisken in 1987 and it was proved that the flow deforms convex initial hypersurface smoothly to a round sphere. This was generalized later by McCoy in 2005 and 2017 to volume preserving flows driven by a large class of 1-homogeneous symmetric curvature functions. In this talk, we will discuss the flows with higher homogeneity and describe the convergence result for volume preserving curvature flows in Euclidean space by arbitrary positive powers of k-th mean curvature for all k=1,...,n. As key ingredients, the monotonicity of a generalized isoperimetric ratio will be used to control the geometry of the evolving hypersurfaces and the curvature measure theory will be used to prove the Hausdorff convergence to a sphere. We also discuss some generalizations including the flows in the anisotropic setting, and the flows in the hyperbolic setting. The talk is based on joint work with Ben Andrews (ANU), Yitao Lei (ANU), Changwei Xiong  (Sichuan Univ.), Bo Yang (CAS) and Tailong Zhou (USTC).