We will consider several classical problems for hypoelliptic diffusions and random walks: the large deviations principle (LDP), the small ball problem (SBP), Chung’s law of iterated logarithm (LIL), and finding the Onsager-Machlup functional. As two very different examples we will look at hypoelliptic Brownian motion and the corresponding random walk on the Heisenberg group, and the Kolmogorov diffusion. We will explore the role of space-time scaling property, Gaussianity, and spectral properties via Dirichlet forms in these settings. The Onsager-Machlup functional is used to describe the dynamics of a continuous stochastic process, and it is closely related to the SBP and LIL, as well as the rate functional in the LDP. Unlike in the elliptic (Riemannian) case we do not rely on the tools from differential geometry such as comparison theorems or curvature bounds as these are not always available in the hypoelliptic (sub-Riemannian) setting. The talk is based on the joint work with Marco Carfagnini, Tai Melcher and Jing Wang.