Neuromorphic (aka brain-inspired) computing is an exciting new paradigm that can make computing dramatically more energy-efficient while potentially offering a path to realize the brain’s elusive artificial intelligence capabilities. As today’s neuromorphic hardware is already delivering an energy-efficient alternative to conventional approaches, the challenge has shifted to the math and computing communities – can we come up with strategies and techniques to take advantage of this potential for scientific computing applications? In our work in neuromorphic algorithms, we have realized that neuromorphic computing may impact a much broader range of applications than widely appreciated, but to do so we must be willing to move beyond the proven recipes that work for conventional computing. In this talk, I will present several vignettes of how we can create new applications for neuromorphic hardware today and going forward. I will first describe the value proposition of neuromorphic systems relative to modern alternatives such as GPUs and CPUs. To illustrate this, I will show how spiking neuromorphic hardware, such as the Intel Loihi platform or the ARM-based SpiNNaker platform, can be used to implement Monte Carlo random walk processes with a wide range of numerical computing applications. Finally, I will introduce a new approach to neuromorphic computing that emphasizes the stochastic nature of the brain that we call COINFLIPS.