The talk presents extensions of the popular bounded difference inequality (also called McDiarmid's inequality) to functions of independent random variables whose conditional deviations are sub-Gaussian or sub_exponential. Applications to machine learning are concentration results for sums of random vectors with sub-exponential norms and a very quick proof of uniform convergence for principal subspace selection, (also called PCA) for vectors with sub-Gaussian norms. If time permits I will also sketch an easy extension of the method of Rademacher complexities to some situations with unbounded input and output variables.