We have done extensive structural and computational studies on the recently discovered phenomenon of negative thermal expansion (NTE) in metal-organic framework (MOF) materials, in particular Cu3(btc)2 and Zn4O(bdc)3 (MOF-5). Through this we have obtained a clear picture of the NTE mechanism as a result of the temperature dependent softening of low energy phonons. We were able to directly observe this on a recent TOSCA experiment on MOF-5 (RB 820198) and propose a similar experiment for Cu3(btc)2. We also observed a dramatic change of the vibrational spectrum of MOF-5 with pressure, which we also propose to investigate in Cu3(btc)2.