The N-branching Brownian motion with selection (N-BBM) is a particle system consisting of N independent particles that diffuse as Brownian motions in R, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. It has associated to it a linearised FKPP equation with a free-boundary (FBPDE). We present the solution of two associated longstanding open problems. Firstly we present the proof, jointly with Julien Berestycki, that the stationary distribution of the N-particle system converges, as N-> infinity, to the travelling wave with minimal wave speed of FBPDE. Secondly we present the proof, jointly with Julien Berestycki and Sarah Penington, that FBPDE displays the same asymptotic behaviour as established by Bramson for the FKPP equation on the whole line. We present extensions of these results to a generalised free-boundary PDE which we show displays pushed and pushmi-pullyu behaviour, depending on a certain parameter. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.