High dimensional data means that the number of variables p if far larger than the number of observations n. This talk starts from a survey of various solutions in linear regression . When p n the OLS estimator does not exist . Since it is a case of forced multi- collinearity, one may use regularized techniques such as ridge regression, principal component regression or PLS regression which keep all the predictors. However if p n combinations of all variables cannot be interpreted. Sparse so- lutions, ie with a large number of zero coe cients, are preferred. Lasso, elastic net, sparse PLS perform simultaneously regularization and variable selection thanks to non quadratic penalties: L1, SCAD etc. In PCA, the singular value decomposition shows that if we regress principal com- ponents onto the input variables, the vector of regression coe cients is equal to the factor loadings. It su ces to adapt sparse regression techniques to get sparse ver- sions of PCA and of PCA with groups of variables. We conclude by a presentation of a sparse version of Multiple Correspondence Analysis and give several applications.