Hipparcos trigonometrical parallaxes of Cepheid variables are used to derive a zero-point for the period-luminosity (PL) relation. Adopting a slope from the Large Magellanic Cloud (LMC), the relation is found to be <M_V>=-2.81 log P-1.43. The standard error of the zero-point is 0.10 mag. Together with metallicity corrections this corresponds to a distance modulus of 18.70+/-0.10 for the LMC and 24.77+/-0.11 for M31. Some implications of these results are discussed. Estimates of the Hubble constant (H_0) that are based on Cepheid observations together with an adopted LMC distance modulus of 18.50 will on average now need to be decreased by ~10 per cent. However, metallicity corrections, which have frequently been ignored, will result in the actual percentage change varying with the sample of galaxies studied. Calibration of RR Lyrae absolute magnitudes using the LMC and M31 Cepheid distances implies an age for the oldest Galactic globular clusters of ~11 Gyr. The parallax data show that the period of Polaris corresponds to first overtone pulsation.