Histograms are convenient and easy-to-use tools for estimating the shape of data, but they have serious problems which are magnified for multivariate data. We combine classic histograms with shape estimation by polynomials. The new relatives, `polynomial histograms', have surprisingly nice mathematical properties, which we will explore in this talk. We also show how they can be used for real data of 10-20 dimensions to analyse and understand the shape of these data.