Let G be a bipartite graph with bipartition (A, B). We give new criteria for a bipartite graph to have an f -factor, a (g, f)-factor and other factors together with some applications of these criteria. These criteria can be considered as direct generalizations of Hall’s marriage theorem. Among some results, we prove that for a function h:A∪B→{0,1,2,…}\documentclass[12pt]{minimal}
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\begin{document}$$h: A\cup B \rightarrow \{0,1,2, \ldots \}$$\end{document}, G has a factor F such that degF(x)=h(x)\documentclass[12pt]{minimal}
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\begin{document}$$\deg _F(x)=h(x)$$\end{document} for x∈A\documentclass[12pt]{minimal}
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\begin{document}$$x\in A$$\end{document} and degH(y)≤h(y)\documentclass[12pt]{minimal}
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\begin{document}$$\deg _H(y) \le h(y)$$\end{document} for y∈B\documentclass[12pt]{minimal}
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\begin{document}$$y\in B$$\end{document} if and only if h(X)≤∑x∈NG(X)min{h(x),eG(x,X)}\documentclass[12pt]{minimal}
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\begin{document}$$h(X) \le \sum _{x\in N_G(X)}\min \{h(x), e_G(x,X)\}$$\end{document} for all X⊆A\documentclass[12pt]{minimal}
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\begin{document}$$X\subseteq A$$\end{document}.
