The main idea of this ECCOMAS MSF 2019 thematic conference is to examine recent advances from mechanics and applied mathematics in a currently very active research domain of multi-scale modeling and computations in solid and fluid mechanics. The latter involves methods which bridge phenomena taking place at multiple scales in space and time and which ought to be placed in interaction or accounted for simultaneously in order to provide the most reliable explanations. This class of problems calls for the development and combination of different analytical tools (homogenization, asymptotic analysis) and computational methods (parallel computing, stochastic analysis) in order to advance the field towards currently relevant nonlinear applications. A number of different schools have developed in various domains of fluids and solids, both in mathematics and mechanics, with sometimes very little or no interaction between them. It is an explicit goal of this thematic conference, in the true spirit of ECCOMAS, to bring these different communities together, and thus provide a sound basis for a fruitful exchange of ideas among them.