Rough Volterra equations 1: the algebraic integration setting

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H\"older exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3.