**Abstract:** Novel mathematics and mathematical modelling approaches together with scalable algorithms are needed to enable key applications at extreme-scale. This is especially true as HPC systems continue to scale up in compute node and processor core count. At the moment computational scientists are at the critical point/threshold of novel mathematics development as well as large-scale algorithm development and re-design and implementation that will affect most of the application areas. Therefore the talk will focus on the mathematical and algorithmic challenges and approaches towards exascale and beyond. It will present ideas and some novel mathematical methods and approaches for extreme-scale systems, such as stochastic and hybrid methods that in turn lead to scalable scientific algorithms with minimal or no global communication, hiding network and memory latency, have very high computation/communication overlap, have no synchronization points. Examples will be drawn from diverse application areas.