Buckling, twisting and fracture are ubiquitous phenomena that, despite having been studied for centuries, still pose many interesting conceptual and practical challenges. In this talk, I will summarize recent theoretical and experimental work that aims to understand the role of curvature and torsion in wrinkle pattern selection, fragmentation cascades and knots. First, we will show how changes in curvature can induce phase transitions and topological defects in the wrinkling patterns on curved elastic surfaces. Thereafter, we will revisit an observation by Feynman who noted that dry spaghetti appears to fragment into at least three (but hardly ever two) pieces when placed under large bending stresses. Using a combination of experiments, simulations and analytical scaling arguments, we will demonstrate how twist can be used to control binary fracture of brittle elastic rods. Finally, in the last part, we will try to shed some light on how topology and torsion affect the stability of knots.