  This article is devoted to the extension of the theory of rough paths in the
context of Volterra equations with possibly singular kernels. We begin to
describe a class of two parameter functions defined on the simplex called
Volterra paths. These paths are used to construct a so-called
Volterra-signature, analogously to the signature used in Lyon's theory of rough
paths. We provide a detailed algebraic and analytic description of this object.
Interestingly, the Volterra signature does not have a multiplicative property
similar to the classical signature, and we introduce an integral product
behaving like a convolution extending the classical tensor product. We show
that this convolution product is well defined for a large class of Volterra
paths, and we provide an analogue of the extension theorem from the theory of
rough paths (which guarantees in particular the existence of a Volterra
signature). Moreover the concept of convolution product is essential in the
construction of Volterra controlled paths, which is the natural class of
processes to be integrated with respect to the driving noise in our situation.
This leads to a rough integral given as a functional of the Volterra signature
and the Volterra controlled paths, combined through the convolution product.
The rough integral is then used in the construction of solutions to Volterra
equations driven by H\"older noises with singular kernels. An example
concerning Brownian noises and a singular kernel is treated
