Solar particle events can give greatly enhanced radiation at aircraft altitudes, but are both difficult to predict and to calculate retrospectively. This enhanced radiation can give significant dose to aircrew and greatly increase the rate of single event effects in avionics. Validation of calculations is required but only very few events have been measured in flight. The CREAM detector on Concorde detected the event of 29 September 1989 and also four periods of enhancement during the events of 19-24 October 1989. Instantaneous rates were enhanced by up to a factor ten compared with quiet-time cosmic rays, while flight-averages were enhanced by up to a factor six. Calculations are described for increases in radiation at aircraft altitudes using solar particle spectra in conjunction with Monte Carlo radiation transport codes. In order to obtain solar particle spectra with sufficient accuracy over the required energy range it is necessary to combine space data with measurements from a wide range of geomagnetically dispersed, ground-level neutron monitors. Such spectra have been obtained for 29 September 1989 and 24 October 1989 and these are used to calculate enhancements that are compared with the data from CREAM on Concorde. The effect of cut-off rigidity suppression by geomagnetic activity is shown to be significant. For the largest event on record on 23 February 1956, there are no space data but there are data from a number of ground-level cosmic-ray detectors. Predictions for all events show very steep dependencies on both latitude and altitude. At high latitude and altitude (17 km) calculated increases with respect to cosmic rays are a factor 70 and 500 respectively for 29 September 1989 and 23 February 1956. The levels of radiation for high latitude, subsonic routes are calculated, using London to Los Angeles as an example, and can exceed 1 mSv, which is significantly higher than for Concorde routes from Europe to New York. The sensitivity of the calculations to spectral fitting, geomagnetic activity and other assumptions demonstrates the requirement for widespread carriage of radiation monitors on aircraft.