Observed data is often contaminated by undiscovered interlopers, leading to biased parameter estimation. Here we present BEAMS (Bayesian Estimation Applied to Multiple Species) which significantly improves on the standard maximum likelihood approach in the case where the probability for each data point being `pure' is known. We discuss the application of BEAMS to future Type Ia supernovae (SNIa) surveys, such as LSST, which are projected to deliver over a million supernovae lightcurves without spectra. The multi-band lightcurves for each candidate will provide a probability of being Ia (pure) but the full sample will be significantly contaminated with other types of supernovae and transients. Given a sample of N supernovae with mean probability, P, of being Ia, BEAMS delivers parameter constraints equal to NP spectroscopically-confirmed SNIa. In addition BEAMS can be simultaneously used to tease apart different families of data and to recover properties of the underlying distributions of those families (e.g. the Type Ibc and II distributions). Hence BEAMS provides a unified classification and parameter estimation methodology which may be useful in a diverse range of problems such as photometric redshift estimation or, indeed, any parameter estimation problem where contamination is an issue.