Some results of Kolchin, Matsuda and Nishioka concerning algebraic curves whose function field is a differential extension of their field of definition, are extended to the case of algebraic surfaces and abelian varieties. Let \(A\subset {\mathbb{C}}^ r\) be open and connected, \(\delta_ i\) the canonical derivations, \(1\leq i\leq r\), Mer(A) the field of meromorphic functions on A. A pair \(K\subset L\) of subfields of Mer(A) containing \({\mathbb{C}}\), such that \(\delta_ i(K)\subset K\), \(\delta_ i(L)\subset L\), is called a differential extension in Mer(A). Let \(A^*\subset A\), \(K^*\subset L^*\) be a differential extension in \(Mer(A^*)\) and such that \(K\subset K^*\), \(L\subset L^*\) are finite extensions, then we say that \(K\subset L\) is embedded in \(K^*\subset L^*\). Now let V be a K-variety and L its function field, \(n=\dim (V)\), W a smooth projective model of \(L\otimes_ K\bar K\) over \(\bar K;\) \(\kappa\), q and \(\alpha\) denote respectively Kodaira dimension, irregularity and Albanese dimension of W. \(K\subset L\) is called an extension of Fuchsian type if \(\delta_ j({\mathcal O}_{V,x})\subset {\mathcal O}_{V,x}\) for every \(x\in V\) and all j. For such an extension we have \(\kappa =-\infty\) if \(n\geq 1\) and \(q=0\) (Theorem 2). An extension \(K\subset L\) is called an extension of Kolchin type if there exists an abelian variety \({\mathbb{C}}^ n/\Lambda\), \(A^*\subset A\) and meromorphic functions \(\beta_ 1,...,\beta_ n\) on A such that \(\delta_ i\beta_ j\subset K\) and L/K is generated by functions of the form \(\psi (\beta_ 1,...,\beta_ n)\) where \(\psi \in Mer({\mathbb{C}}^ n)\) is abelian with respect to \(\Lambda\)- these extensions play an important rôle in Kolchin’s theory of differential fields. Theorem 1 gives a number of sufficient conditions for an extension of Fuchsian type to be embeddable in an extension of Kolchin type, namely (a) \(n\leq 2,\) \(\kappa\neq - \infty\), \((b)\quad n=3,\) \(\kappa =0\), (c) \(n=3,\) \(\kappa\geq 1\), \(\alpha\neq 3\), (d) L is the function field of an abelian K-variety. A sufficient condition for a differential extension \(K\subset L\), where L is the function field of an abelian K-variety, to be an extension of Fuchsian type is the following: The group of differential K-automorphisms of L is not finite.