This application-oriented work concerns the design of efficient, robust and reliable algorithms for the numerical simulation of multiscale phenomena. To this end, various modern techniques from scattered data modelling, such as splines over triangulations and radial basis functions, are combined with customized adaptive strategies. The resulting multiresolution methods are thinning algorithms, multilevel approximation schemes, and meshfree discretizations for transport equations. The utility of the algorithmic approach taken in this research is supported by the wide range of applications, including image compression, hierarchical surface visualization, and multiscale flow simulation. Special emphasis is placed on comparisons between the various numerical algorithms developed in this work and comparable state-of-the-art methods.