In this long paper, the author studies local systems on smooth (not necessarily complete) curves \(X\) by working in the framework of analytic étale topology of V. Berkovich. The base field \(k\) is a field of characteristic \(0\) complete with respect to a non-archimedean metric and with a residue field of positive characteristic. The notion of an analytic local fundamental group at a point of \(X\) is defined so that a locally constant sheaf on a small punctured disc at the point gives a continuous representation of this group. A canonical quotient of this group is constructed which is the analogue of the local differential Galois group classifying connections with poles of finite orders. An analytic Swan conductor is defined in terms of a canonical higher ramification filtration on the local fundamental group. It is proved that the Swan conductor of a finite rank representation is an integer (an analogue of the Hasse-Arf theorem). The author defines and studies meromorphically ramified local systems on an open curve with certain bounds on the ramification at the points at infinity. The main tool is the Fourier transform. It is shown that the cohomology of a meromorphically ramified local system has finite rank. A formula of Grothendieck-Ogg-Shafarevich type is proved for all meromrphically ramified sheaves on any smooth open curve using a local Morse-theoretic argument. Finally the author defines a Galois module \(\Gamma (q)\) of a quadratic form \(q\) which descends to a homomorphism from the Witt group of \(k\) to the group of isomorphism classes of rank \(1\) \( \ell-\)adic Galois modules. This is inspired by classical work of A. Weil.