Nonlinear ordinary and partial differential equations have been central to modelling of some of the most significant problems in physics, chemistry, engineering, biology and finance, including climate modelling, aircraft design, molecular dynamics and drug design, deep learning neural networks and financial markets. While quantum computation has shown its advantages for solving linear problems, nonlinear problems have yet remained elusive. This is because quantum mechanics itself is fundamentally linear (as far as we know) and it is not yet known how to model nonlinear problems in a linear way without significant approximations that no longer capture truly nonlinear behavior. Truly nonlinear behavior, however, is what makes real physical systems, like the weather and stock markets, interesting, complex and unpredictable. We show that an important class of nonlinear partial differential equations - Hamilton-Jacobi and scalar hyperbolic equations - can indeed be fully captured using a quantum algorithm. These equations are important for many applications like optimal control, machine learning, semi-classical limit of Schrodinger equations, mean-field games and many more. Physical quantities like density and energy (and many more) can be computed using a quantum algorithm that can be up to exponentially more efficient compared to a classical device with respect to the dimension of the system and the error of the final answer. In addition, we provide methods for more general classes of nonlinear partial differential equations and show when speedup with respect to the number of initial conditions can be achieved.