There are two basic approaches to measuring the rate of expansion of the universe:
a) using Cepheid variable stars – their brightness tells us how far away they are, and their redshift facilitates calculating the expansion rate
b) deductions from ripples in the CMB (Cosmic Microwave Background) – originating from the big-bang
These two methods have become more refined over time, but their measured values are some way apart – “about four times the size of their combined uncertainty.” This is according to research published in the Astrophysics Journal, July 12th.
The simplest explanation would be that the expansion rate is greater in the nearby universe than the more distant universe – but such a conclusion is incompatible with the current model, and would require new physics. Research is ongoing.
Update: 8th February 2019: A new technique has been described for measuring the rate of expansion of the Universe (the Hubble constant). Observing the duplicate images of quasars that show when an intervening galaxy acts as a gravitational lens, astronomers can use the slight differences in light path for the images, (which are revealed by the slight difference in arrival-time of brightness variations), to measure the distances to the lens-galaxy and quasar. Combining this with the red-shift gives a measurement of the Hubble Constant – in this case 72.5. This agrees with the Cepheid variable measurements ((a) above) but is 8% higher than measurements form the ancient universe i.e. the CMB measurements. This goes some way to confirming that the rate of expansion is increasing, but does not provide any explanation why.
Update 9th April 2020: A study has just been published showing measurements that indicate the rate of expansion of the universe is not the same in all directions or all places. This ties in with the variations described above in that it explains the different results obtained. It is something of a shock to cosmologists who have always assumed the axiom of isotropy – that the universe is the same whatever direction you look in, which seems to be born out by the CMB. The reason may be that there is more dark energy in some regions than others, or there may be more sophisticated explanations, or there may be a subtle reason why these measurements appear the way they do. Either way we have not heard the end of this – this is a major shock to cosmology.