The aim of this thesis is the study of two different models of spatially extended branching systems. First, we consider a one-dimensional branching Brownian motion that evolves in a spatially random environment. We argue that the quenched fluctuations of the maximally displaced particle re-centred at its median remain bounded in time. For the standard branching Brownian motion in a homogeneous environment, an analogous result already follows by the fact that the distribution function of the re-centred maximally displaced particle corresponds to the critical travelling wave solution of the related Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation. This argument, however, cannot be extended to the inhomogeneous setting. In order to achieve our result, we employ certain tilted path-measures in order to get fine control on the Feynman-Kac representations to solutions of the F-KPP equation, which we combine with an analytic result on the evolution of the number ``zero-crossings'' of solutions to parabolic equations, known as a Sturmian principle. The second model we consider is a discrete-time model of branching annihilating random walk on $\mathbb{Z}^d$. In this model, at the end of each generation, all particles produce a mean $\mu$ number of offspring that disperse uniformly within a fixed distance $R$ from their parent. Whenever two (or more) child particles try to occupy the same site they get annihilated. This local interaction of particles in the branching system has the interesting but also challenging consequence that high local density of particles leads to more annihilation, making the system non-monotone. We investigate and determine regimes of the model parameters for which the system either dies out almost surely or survives with positive probability. Moreover, we exhibit regimes where there is a unique non-trivial ergodic equilibrium distribution that has exponential decay of correlations. Lastly, by keeping track of genealogical information (i.e. parent-child relations), we examine the ancestral ...