Compositional data represent the relative information between variables that are parts of some whole. The relevant information is contained only in the ratios between the measured variables, and not in the absolute values. A common procedure how to analyze this relative information is to use the so-called log-ratio approach, proposed by John Aitchison in the 1980s. From a geometrical point of view, the compositions live in the simplex sample space, and the log-ratio approach enables a representation in terms of coordinates in the usual Euclidean geometry. The well known coordinates are the additive log-ratio (alr), the centered log-ratio, and the isometric log-ratio (ilr) coordinates. The clr and ilr coordinates are preferred, since the ilr representation constructs orthonormal coordinates and the clr representation allows for an interpretation in terms of the original variables. We focus on different aspects of compositional data: One field of interest are high-dimensional compositional data, where the interpretation of the resulting coordinates can become a complex task. Another concern is the propagation of measurement errors in the construction of the orthonormal coordinates. Applications in geochemistry, but also in epidemiology, which is a new field for this kind of analysis, underline the usefulness of this approach.