This books describes a new tool for analyzing discrete data sets. This tool, the Delaunay Tessellation Field Estimator (DTFE), can be used to reconstruct density or intensity fields from discrete point samplings at an optimal resolution. It automatically adapts to both the local density and geometry of the discrete point set. In the first half of this book the DTFE is extensively described and tested. In the second half several astronomical applications are described, such as the analysis of numerical simulations of cosmic structure formation, the mapping of the large scale galaxy distribution in the universe and improving numerical simulation algorithms of cosmic structure formation. The DTFE forms a major step forwards for studies of the large scale galaxy distribution, one of the most exciting research topics in present-day cosmology.