A new definition of hypergeometric functions (HF) is given. One considers the manifold \(G^ 0_{n,N}\) of the \(n\)-dimensional subspaces in \(\mathbb{C}^ N\) containing the vector \((1, 1,\dots,1)\) and the vector bundle \(U_{n,N}\) over \(G^ 0_{n,N}\) (that is dual to the tautological vector bundle). The general hypergeometric system (GHS) associated with the pair \((n,N)\), \(n>N\), of natural numbers is introduced. The holomorphic solutions of this system are called general hypergeometric functions (GHF). The relationship between the GHF and HF is established and the solutions of GHS are obtained as series of hypergeometric type \((\Gamma\)-series). Necessary and sufficient conditions for the functions on \(U_{n,N} \times Y\) to be GHF on \(U_{n,N} \times Y\) are also given. \((Y\) is a complex submanifold in \(\mathbb{C}^ N)\).