Scatterplots are well established means of visualizing discrete data values with two data variables as a collection of discrete points. We aim at generalizing the concept of scatterplots to the visualization of spatially continuous input data by a continuous and dense plot. An example of a continuous input field is data defined on an n-D spatial grid with respective interpolation or reconstruction of in-between values. We propose a rigorous, accurate, and generic mathematical model of continuous scatterplots that considers an arbitrary density defined on an input field on an n-D domain and that maps this density to m-D scatterplots. Special cases are derived from this generic model and discussed in detail: scatterplots where the n-D spatial domain and the m-D data attribute domain have identical dimension, 1-D scatterplots as a way to define continuous histograms, and 2-D scatterplots of data on 3-D spatial grids. We show how continuous histograms are related to traditional discrete histograms and to the histograms of isosurface statistics. Based on the mathematical model of continuous scatterplots, respective visualization algorithms are derived, in particular for 2-D scatterplots of data from 3-D tetrahedral grids. For several visualization tasks, we show the applicability of continuous scatterplots. Since continuous scatterplots do not only sample data at grid points but interpolate data values within cells, a dense and complete visualization of the data set is achieved that scales well with increasing data set size. Especially for irregular grids with varying cell size, improved results are obtained when compared to conventional scatterplots. Therefore, continuous scatterplots are a suitable extension of a statistics visualization technique to be applied to typical data from scientific computation.