Isometric embedding of Riemannian manifolds into Euclidean space is a classical problem in differential geometry. In this talk, after introducing some literature on such topic, I will present my recent works on isometric embeddings of different regularity. First I will show global C^{1, \theta} Nash-Kuiper theorems for compact manifolds with sharper Holder exponent, which is about the flexibility of isometric embedding, through the powerful technique of convex integration.  Then I will also show my Ph. D works on C^{1, 1} isometric immersions of two types of metric by compensated compactness theory and global smooth isometric immersions of negatively curved surfaces with slowly decaying curvature through the characteristic method.