A general description of various path-dependent functions is given by using series expansions of Taylor type. Path-integrals and path-tensors are defined as many-components quantities whose values contain complete information on the path. These quantities are considered as elementary path-dependent functions and are used instead of power monomials in the usual Taylor series. The coefficients of such an expansion are interpreted as partial derivatives dependent on the order of the differentiations or else as non-standard covariant two-point derivatives. Some examples of path-dependent functions are given: the work function in a non-potential field \(W(x',x)=\int^{x'}_{x}F_{\nu} dx^{\nu}\), the parallel-transport tensor \(\Theta^{\alpha}_{\beta}(x',x): V^{\beta}\to V^{'\alpha}=\Theta^{\alpha}_{\beta} \quad V^{\beta},\) the function \(\Theta^{e\mu}(x',x)=-g^{\mu \nu}\cdot \partial_{\nu}\Omega (x',x),\) where \(\Omega\) (x',w) is the world Synge function. The curvature tensor is considered and its properties are determined under a non-transitive translator of a general type. The extension to general tensor fields is also considered.