This talk is based on my upcoming book chapter of the same name, to appear in Springer handbook of the mathematics of the arts and sciences. I will sample the myriad of ways in which mathematical experiment advances my research. The examples are problems I solved with techniques inspired by the methodology and writings of the late Jonathan Borwein. I will emphasize the tools, strategies, and the broader arc that research follows: from low-dimensional, specific, and visible, to high-dimensional and general. Topics include dynamical systems, geometry, optimization, error bounds, random walks, special functions, and number theory. Because I am mainly interested in tools and creative thinking, rather than specific theory, this talk should be accessible to a wide audience.