Fluctuations are ubiquitous in nature and are relevant for nearly every ecological process. The main sources of fluctuations in population abundances are demographic and environmental stochasticity, whose effect on local population dynamics, metapopulations and metacommunities have attracted much interest in the ecological literature. A third source of stochasticity is demographic heterogeneity, which is the variability of demographic traits within a population. Despite the large body of literature dedicated to fluctuations in ecology, their role in some relevant ecological patterns and processes is still rather unexplored. For example, the effect of demographic and environmental stochasticity on species spread is poorly understood, mostly due to a scarcity of experimentation linking theoretical models with replicated experiments. Additionally, environmental stochasticity can induce population fluctuations and has been shown theoretically to determine the exponent of one of the most widespread scaling laws in nature, Taylor's law of fluctuation scaling. However, empirical observations point towards the existence of a single universal Taylor’s law exponent, in contrast with such model predictions. Here, experiments with protist microcosms and methods from statistical physics are used to investigate the role of fluctuations and heterogeneity on relevant ecological patterns and processes. The effect of demographic and environmental stochasticity on the propagation of biological invasions is studied in microcosm experiments with Tetrahymena sp. and Euglena gracilis and with stochastic generalizations of the Fisher-Kolmogorov equation. Demographic stochasticity is shown to induce fluctuations in the position of the propagating front and the statistical structure of the environmental heterogeneity is shown to cause a slowing-down of the invasion front at large autocorrelation lengths. The investigation of biological invasions in environments with heterogeneous distribution of resources is performed experimentally by manipulating light, the energy resource for photosynthetic organisms. Such experimental setup is further used to study phototaxis, the directed motion of phytoplankton towards or against light sources, a process that is important for relevant ecological phenomena such as diel vertical migration. A model for phototaxis is derived from the experiments in the generalized Keller-Segel framework. Large deviations theory is used to derive a generalized Taylor's law and to elucidate the origin of a universal scaling exponent as due to sampling rather than to the population growth process. The framework of finite-size scaling is used to characterize the demographic heterogeneity in a relevant ecological trait, the body size of individuals. Intra-specific body size distributions measured experimentally are shown to be described by a universal scaling distribution across different taxa and over four orders of magnitude in body size. Mathematical models of cell growth and division are shown to be compatible with the observed universal body size distribution.