Real world data is frequently given via a continuous function relating two or more quantities of interest with each other. However, for various reasons we might only have a finite number of samples to work with at any time. This leads to the question how one can extract as much information as possible from a given number of sample nodes. In this thesis we look at such problems from a geometric, harmonic analytic and approximation theoretic point of view. We start by giving a comprehensive overview on uniform point sets (in the sense of low discrepancy, quasi-Monte Carlo worst case errors and energy minimizers) as well as an extensive account of techniques and known results in this area. Throughout the thesis we focus on point sets in the cube and torus, first showing that certain combinatorially defined point sets connect the periodic structure of the torus with the nonperiodic one of the cube. We then look at the problem of energy minimizers in the torus in a product sense, in particular analyzing the optimality of so called Fibonacci lattices for this purpose. To conclude we turn to the question of whether one can give simple constructions of optimal point sets for the quasi-Monte Carlo method in the cube in any dimension, particularly arguing for the use of tent transformed order 2 nets in this context.
