Fractals are patterns that repeat at many magnifications. These intricate patterns are found throughout nature, ranging from clouds, rivers and lightning through to our brains, blood vessels and lungs. Due to their prevalence in nature and their growing impact on technology, fractals have assumed a rapidly expanding role across the sciences and arts. In this talk, I will explore some of the intriguing properties of fractals by taking a scientific journey through the research fields I have worked in. These include nano-electronic circuits, artificial retinas, solar cells, collapsing Antarctic ice-shelves, neural brain structure and the stress-reducing properties of artworks. I hope to show a common theme – that quantification of their underlying fractal geometry provides an enhanced understanding well beyond the traditional qualitative views of these diverse systems.