A new mathematical representation for nonlinear viscoelasticity is presented based on the application of the Volterra series expansion to the general nonlinear relationship between shear stress and shear strain history. This theoretical and experimental framework, which we call medium amplitude parallel superposition (MAPS) rheology, reveals a new material property, the third-order complex modulus, which describes completely the weakly nonlinear response of a viscoelastic material in an arbitrary simple shear flow. In this first part, we discuss several theoretical aspects of this mathematical formulation and new material property. For example, we show how MAPS measurements can be performed in strain- or stress-controlled contexts and provide relationships between the weakly nonlinear response functions measured in each case. We show that the MAPS response function is a superset of the response functions that have been previously reported in medium amplitude oscillatory shear and parallel superposition rheology experiments. We also show how to exploit inherent symmetries of the MAPS response function to reduce it to a minimal domain for straightforward measurement and visualization. We compute this material property for a few constitutive models to illustrate the potential richness of the datasets generated by MAPS experiments. Finally, we discuss the MAPS framework in the context of some other nonlinear, time-dependent rheological probes and explain how the MAPS methodology has a distinct advantage over these others because it generates data embedded in a very high-dimensional space without driving fluid mechanical instabilities, and is agnostic to the flow protocol. (c) 2020 The Society of Rheology.