I will discuss some recent progress concerning the distribution of quadratic Weyl sums. These sums are classical objects in number theory and also appear in quantum mechanics. Analogously to Donsker's theorem in probablity theory, the existence of limiting distribution for the interpolated partial sums (weak invariance principle) allows us to define a limiting random process. This process has several non-standard properties and is related to the geodesic flow on a noncompact hyperbolic manifold. Time permitting, I will also discuss a stronger invariance principle, which allows us to uniformly approximate any quadratic Weyl sum. This fact relates to classical results by Hardy and Littlewood, Mordell, and to more recent results by Fedotov and Klopp. Joint work with Jens Marklof (Bristol)