Solving a System of Nonlinear Equations

In any normal linear algebra class you learn how to solve a system of linear equations using Gaussian elimination methods. But how would solve a system like the following:
$$x^2+y+z=1$$

$$x+y^2+z=1$$

$$x+y+z^2=1$$

In this talk I will go over how one can use Gr\"obner theory to solve a system of nonlinear equations and what assumptions might need to be made to make this possible.