Continuous physical processes pervade every aspect of our society, industry and the natural world. From the flow of air over an aircraft to the propagation of mobile phone signals, to the behaviour of chemical components at every point of the manufacturing processes, continuum mechanics is at the heart of our industrial processes. In medicine, the electrical behaviour of the heart and brain, the flow of blood and other fluids through the body, and the detection of disorders using all manner of scanners and detectors are all continuum mechanics processes. In the natural world, detecting and understanding the movement and composition of the Earth enable us to understand earthquakes and to hunt for valuable minerals, while advanced understanding of the complex interaction of fluids and electromagnetic fields allows us to understand stars, the cosmos and our place in it. In all of these cases and many more beside, the mathematical equations describing phenomena are known, but solutions very rarely exist. Science and engineering are essentially dependent on computer simulation to understand any of these systems, and to design the devices and processes which use them. Many of these phenomena are so complex or have such a range of spatial scales that existing petascale computer systems are a limit on scientific advance. In addition, there is a need to go beyond mere simulation to simulate the uncertainty in processes, find the optimal solution, or discover the multiple possible outcomes of a system. The advent of exascale computing presents the opportunity to address these limitations. However, increasing computational scale, increasingly complex simulation algorithms, and the vast quantities of data produced by exascale computing will defeat not just existing simulation software, but also existing ways of writing simulation software. Gen X is a project to establish the requirements for exascale simulation software for continuum mechanics, and to provide a concrete way of achieving this capability within the next five years. The Gen X approach is to move beyond just writing code to a system of specialist simulation languages which enable scientists and engineers to specify the problem they want to solve and the algorithms they want by writing mathematics, the language of science. The actual code will be automatically generated by specialist compilers rather than hand-written. Rather than an algorithm developer writing a paper about their new development and hoping that simulation scientists will find the time to code it up for their specific problem, the algorithm will be encoded in a domain specific language and implemented in its compiler. The simulation scientist will then be able to access the algorithm directly without recoding. At exascale, writing all the simulation outputs to disk for later analysis is impossible. Instead, simulation data must be processed, analysed and visualised as the simulation is conducted, and only the results stored for later use. Gen X will provide mathematical languages for this process which will enable the scientist or engineer to concisely specify the analysis to be performed, and to have confidence that the resulting calculations will be both efficient and correct. By enabling scientists and engineers to work at a higher mathematical level while also accessing more sophisticated algorithms and hardware-specific implementations than previously possible, Gen X will make simulation science both more capable and more productive. In this manner, Gen X is essential to realising the potential of exascale computing while also making the most efficient use of research resources.