We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x > 0$, particles move according to independent one-dimensional Brownian motions with the critical drift of negative the square root of 2, and particles are absorbed when they reach zero. Kesten (1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time t. These bounds improve upon results of Kesten (1978), and partially confirm nonrigorous predictions of Derrida and Simon (2007). We will also discuss results concerning the behavior of the process before the extinction time, as x tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle, and we obtain asymptotic results for the configuration of particles at a typical time. This is based on joint work with Julien Berestycki and Nathanael Berestycki.