We consider the problem of Gaussian mixture model (GMM)-based classification of noisy data, where the uncertainty over the data is given by a Gaussian distribution. While this uncertainty is commonly exploited at the decoding stage via uncertainty decoding, it has not been exploited at the training stage so far. We introduce a new Expectation-Maximization (EM) algorithm called uncertainty training that allows to learn GMMs directly from noisy data while taking their uncertainty into account. We evaluate its potential for a speaker recognition task over speech data corrupted by real-world domestic background noise, using a state-of-the-art signal enhancement technique and various uncertainty estimation techniques as a front-end. Compared to conventional training, the proposed algorithm results in 3\% to 4\% absolute improvement in speaker recognition accuracy by training from either matched, unmatched or multi-condition noisy data. This algorithm is also applicable with minor modifications to maximum a posteriori (MAP) or maximum likelihood linear regression (MLLR) model adaptation and to the training of hidden Markov models (HMMs) from noisy data.