Branching Brownian motion (BBM) is a random particle system which incorporates both the tree-like structure and the diffusion process. In this talk, we will consider two variant models of BBM, BBM with absorption and BBM with an inhomogeneous branching rate. In the first model, we will study the transition from the slightly subcritical regime to the critical regime and obtain a Yaglom type asymptotic result of the expected number of particles conditioned on survival as the process gets closer to being critical. In the second model, we will see how it can be used to study the evolution of populations undergoing selection. We will provide a mathematically rigorous justification for the biological observation that the distribution of the fitness levels of individuals in a population evolves over time like a traveling wave with a profile defined by the Airy function. The second part of the talk is based on joint work with Jason Schweinsberg.