In this thesis, I describe a range of scenarios, from viral propagation to stem cell dynamics, where techniques and ideas from statistical physics provide a route to describe and quantify the observed dynamics. All the systems considered here lie in the realm of out-of-equilibrium physics, where injection and dissipation of energy, matter and/or momentum drive their spatio-temporal evolution. This thesis is organised such that the topics are presented from mainly theoretical to mainly experimental. In Chapter 2, I make use of field-theoretic methods to study a branching random walk. This is a paradigmatic process in the study of viral propagation, however analytic results are very limited. Here I show how a field-theoretic approach provides an a route to obtain exact results for the scaling of the volume explored by such a process. In Chapter 3, I show how branching or self-replication can emerge in large scale ecological systems. I show, numerically, how a spatial instability of the vegetation patches gives rise to their self-replication, and discuss the implications for real ecosystems. In Chapter 4, I dive into the realm of cellular biology, where I performed experimental, analytical and numerical work in order to understand the rich dynamics of the spatio-temporal interactions of mouse embryonic stem cells and localized sources of protein signals, and discuss the implications for multicellular organisation. In Chapter 5, I discuss my work on human Keratinocytes, where I studied the interplay between the pulsatile activity of a specific pathway and differentiation. I introduce a method that allows the construction of a phase diagram from the stem cell state. Combined with numerical simulations, this method allowed the visualisation of the temporal relation between the signals to show how transitions between stem cell states occur. Finally, in Chapter 6, I discuss the main findings of this thesis. I present the general and specific conclusions and point out some key open problems on each sub-field studied.