Alkali-Transition metal Oxides and polyanionic compounds exhibit complex and unique properties such as redox activity associated to alkali diffusion for Li battery materials and peculiar electronic properties for thermo-electricity generation, two domains for which the demand for improved comprehension and devices is currently critical. The current research is extremely active and competitive at the international level in the two domains, mostly based on characterization at the macro-scale such as diffraction, as well as electric, magnetic, and thermal properties. The required properties are governed by 'Electronic aspects (localization-delocalization of electrons; spin state; metallic vs. paramagnetic behaviour etc.) induced by the electrons on the d orbitals of the transition metal(s) 'Ionic aspects (ordering; mobility; coupling with electrons etc.) induced by the alkali ion(s). Both aspects are determined of course by the atomic structure, the knowledge of which is often quite advanced through global-scale techniques as X-ray, neutron and electron diffraction, but the key for the understanding actually resides in the very local scale (defects, partial ordering, electron localization, ion-electron coupling etc). A particularly relevant method for approaching the local structure is NMR spectroscopy. It is indeed sensitive to the atomic, electronic and magnetic local environment of a large number of nuclei, many of which are quite common in the materials of interest (Li, Na, Co, P, O F etc.). NMR is also sensitive to dynamic effects in a rather wide range of time scales (from tens of MHz to Hz). However, the richness of the information carried by NMR is often obscured by the complexity of the interactions that govern the signal in such complex materials, and the lack of understanding of some of these interactions, notably the so-called hyperfine ones, exerted by delocalized or single electrons onto the nuclei probed by NMR. Improvement of the understanding of such interactions is therefore a key to the full exploitation of NMR in these systems. Our proposal consists in a strategy including the selection of adequate compounds, their extensive global-scale characterization (diffraction, magnetism, conduction), their high resolution (Magic Angle Spinning) NMR characterization (under varying temperature conditions when appropriate and possible) and the development of the calculation methods suitable to shed light on the mechanisms of the interactions that govern the spectrum possibly in presence of movement. The progress thus expected in the understanding of these interactions will then be applied to more complex compounds, those that are at the forefront of the application-driven research in the fields of Li battery and of thermoelectric materials. In direct relation with the critical properties of the compounds investigated, our efforts will be devoted to two aspects of the modelling of NMR spectra: (i) The Fermi contact shift interaction, and (ii) the lineshape alterations due to movement. (i) Our strategy to interpret the NMR measurements for paramagnetic battery materials is to improve previous established DFT modelling based on the topology of the spin density around the lithium site using pseudo-potentials with a plane-waves basis set. Reconstruction of the 'true' wave function from the pseudo-one is needed and the PAW(25) (projector augmented wave) implementation will be used for this purpose. However, the reconstruction needs to be validated using an all electron method like the WIEN2k package. By using an all electron/full potential approach, we will also be able to reach a better accuracy for the hyperfine field as fully relativistic core wave functions can be used (this can have an important effect very close to the nucleus), together with various magnetic ordering schemes, including non-collinear magnetism coupled with spin-orbit coupling. Once the calculation of the hyperfine field will be established using WIEN2k or a PAW implementation, the coupling with the NMR paramagnetic chemical shift will be done using magnetic susceptibility measurements as a function of the temperature. (ii) In the context of ion mobility, the exploitation of the quadrupolar lineshape like that of 23Na in particular is crucial. The calculation model proposed by J. H. Kristensen and I. Farnan in 2001 to simulate chemical exchange will be adapted to the kind of compounds of interest. The model relies on the simulation of the lineshape based on the calculation of the electric field gradient (in the presence of electron spins) and on hopping schemes between crystallographic sites. Close cooperation of the Rennes partner with the initiator of this model will ensure that the step of the paramagnetic compounds will be successfully passed. The three laboratories involved in the project have complementary expertise and a solid internationally recognized experience in the fields of interest.