To any field $k$, there is a motivic stable homotopy category of schemes over $k$. In this setting, one can construct a motivic Adams spectral sequence (MASS) which converges to something related to the homotopy groups of spheres. This talk will introduce the motivic stable homotopy category and show how the MASS over the complex numbers relates to the classical Adams spectral sequence. I will then discuss joint work with Paul Ostvaer on the MASS over finite fields.