We review recent progress in describing the statistics of height fluctuations in 1D Kardar Parisi Zhang (KPZ) growth, focusing on the KPZ equation and its integrability properties via the mapping onto the Lieb Liniger model of impenetrable bosons. We recall the replica Bethe Ansatz method and how it allows to calculate one time probabilities, and shows the emergence of the Tracy Widom distributions of random matrix theory. We then study the two-time problem, the so called aging problem, which is still outstanding: the aim is to obtain the joint probability distribution of heights at time t and t', in the limit of large times with fixed ratio t/t' > 1. We provide a partial solution of this problem, exact in some limits. In particular we derive the exact form of the persistent correlations in the limit t/t' large, which quantifies the memory effect in the time evolution, also called ergodicity breaking. Comparison with experiments and numerics shows a very nice agreement. Most is joint work with J. de Nardis and K. Takeuchi.