We describe a statistical technique, which we call Moment Component Analysis (MCA), that extends principal component analysis (PCA) to higher co-moments such as co-skewness and co-kurtosis. This method allows us to identify the factors that drive co-skewness and co-kurtosis structures across a large set of series. We illustrate MCA using 44 international stock markets sampled at weekly frequency from 1994 to 2014. We find that both the co-skewness and the co-kurtosis structures can be summarized with a small number of factors. Using a rolling window approach, we show that these co-moments convey useful information about market returns, for systemic risk measurement and portfolio allocation, complementary to the information extracted from a standard PCA or from an independent component analysis.