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We classify finite-dimensional pointed Hopf algebras with abelian coradical,up to isomorphism, and show that they are cocycle deformations of theassociated graded Hopf algebra.  More generally, for any braided vector space of diagonal type $V$ with aprincipal realization in the category of Yetter-Drinfeld modules of acosemisimple Hopf algebra $H$ and such that the Nichols algebra$\mathfrak{B}(V)$ is finite-dimensional, thus presented by a finite set$\mathcal G$ of relations, we define a family of Hopf algebras$\mathfrak{u}(\boldsymbol\lambda)$, $\boldsymbol\lambda\in \Bbbk^{\mathcal G}$,which are cocycle deformations of $\mathfrak{B}(V)\# H$ and such that$\operatorname{gr}\mathfrak{u}(\boldsymbol\lambda)\simeq \mathfrak{B}(V)\# H$.

Liftings of Nichols algebras of diagonal type II. All liftings are cocycle deformations