Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function $P(x,t)$ of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells, and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the large deviations approach for a continuous time random walk, we uncover a general universal behavior for the decay of the density . It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to exponential decay (with logarithmic corrections) of $P(x,t)$. This universal behavior also holds for short times, a fact that makes experimental observations readily achievable. Time permitting, we will discuss the Hitchhiker model  which gives a microscopical description of the observed behavior, in terms of a model describing aggregation processes, that leads to a distribution of molecule sizes and to spatial exponential decay of the spreading particles.
 E, Barkai, Stas Burov, Packets of diffusing particles exhibit universal exponential tails, Physical Review Letters 124, 060603 (2020).
 M. Hidalgo-Soria, and E. Barkai The Hitchhiker model for Laplace diffusion processes in the cell environment, Physical Review E 102, 012109 (2020).