Muon beams of low emittance provide the basis for the intense, well-characterised neutrino beams necessary to elucidate the physics of flavour at the Neutrino Factory and to provide lepton-anti-lepton collisions at energies of up to several TeV at the Muon Collider. The International Muon Ionization Cooling Experiment (MICE) will demonstrate ionization cooling; the technique by which it is proposed to reduce the phase-space volume occupied by the muon beam at such facilities. In an ionization-cooling channel, the muon beam is caused to pass through a material (the absorber) in which it looses energy, the energy lost is then replaced using RF cavities. The combined effect of energy loss and re-acceleration is to reduce the transverse emittance of the beam (transverse cooling).MICE is being constructed in a series of Steps. At Step IV, MICE will be able to study the properties of liquid hydrogen and lithium hydride that affect cooling. A solenoidal spectrometer will measure emittance upstream and downstream of the absorber vessel. The muon beam will be focused at the absorber by a focusing coil. The construction of Step IV at the Rutherford Appleton Laboratory is well advanced and is scheduled to be complete early in 2015. The status of the construction project will be described together with the performance of the principal components. Once the Step IV programme has been completed, the apparatus will be reconfigured to allow the MICE collaboration to demonstrate ionization cooling. This will require two single-cavity modules to be inserted one upstream and one downstream of a central absorber. The status of the preparations for the MICE demonstration of ionization cooling will also be described briefly.