This paper presents empirical evidence on the nature of idiosyncratic shocks to firms and discusses its role for firm behavior and aggregate fluctuations. We document that firm-level sales and productivity are hit by heavy-tailed shocks and follow a nonlinear stochastic process, thus departing from the canonical linear. We estimate a state-of-the-art model to flexibly capture the rich dynamics uncovered in the data and characterize the drivers of nonlinear persistence and non-Gaussian shocks. We show that these features are crucial to get empirically plausible volatility and persistence of micro-originated (granular) aggregate fluctuations.