The aim of this paper is to overview challenges and principles of Big Data analysis in biomedicine. Recent multivariate statistical approaches to complexity reduction represent a useful (and often irreplaceable) methodology allowing performing a reliable Big Data analysis. Attention is paid to principal component analysis, partial least squares, and variable selection based on maximizing conditional entropy. Some important problems as well as ideas of complexity reduction are illustrated on examples from biomedical research tasks. These include high-dimensional data in the form of facial images or gene expression measurements from a cardiovascular genetic study.