Final Activity Report Summary - TDRMX (Time-dependent R-matrix) The development and implementation of an ab-initio theoretical approach for the handling of the interaction of atomic and molecular systems with strong and ultra-short electromagnetic (EM) fields has been the subject of the present project. The method is developed in detail and applied in the case of hydrogen. The complete theoretical framework which describes the full linear and non-linear dynamics for the case of multi-electron systems has been developed as a basis for the implementation of the method in these complex systems. In terms of the EM field characteristics the applicability of the theory ranges from the far infrared (fraction of eV) to x-ray (Kev) spectral band and pulse durations from attoseconds to hundreds of femtoseconds (1 femto-second is a millionth-billionth of a second and 1 atto-second is a thousand times shorter still). In addition, a novel numerical approach has also been developed which consists in the merging of two major numerical algorithms in solving a multi-dimensional partial differential equation (PDE). To the best of our knowledge this numerical approach is the first of its kind, at least in the area of theoretical Atomic, Molecular and Optical (AMO) physics.