We discuss the concept of distributional alchemy. This is defined by “transmutation” maps that are the functional composition of the cumulative distribution function of one distribution with the inverse cumulative distribution (quantile) function of another. We show that such maps can lead on the one hand to new and tractable methods for the introduction of skewness or kurtosis into a symmetric or other distribution, without the pathology of Gram-Charlier expansions, and on the other hand to practical methods for converting samples from one distribution into those from another, without the limitations of Cornish-Fisher expansions. These maps have many applications in statistics generally and mathematical finance in particular, including the assessment of distributional risk in pricing and risk calculations. We give examples of skew-uniform, skewnormal and skew-exponential distributions based on these techniques, suggest kurtotic variations, and also describe accurate methods for converting samples from the normal distribution into samples from the Student distributions or for converting one Student distribution into another.