Non-smooth saddle-node bifurcations of one-parameter families of quasiperiodically driven dynamical systems on the real line give rise to strange non-chaotic attractors. In this thesis, we provide a class of families which has non-empty interior in the C2-topology and whose elements undergo non-smooth saddle-node bifurcations. Within this class, we study the geometry of the corresponding strange attractors by computing different fractal dimensions. In particular, we show that the Hausdorff dimension differs from the box-counting dimension. We further obtain a description of the minimal set at the bifurcation as a maximal invariant set. Our results treat both the discrete and the continuous time case. A number of explicit examples emphasise the applicability of our findings.