Montage is a portable software toolkit for constructing custom, science-grade mosaics by composing multiple astronomical images. The mosaics constructed by Montage preserve the astrometry (position) and photometry (intensity) of the sources in the input images. The mosaic to be constructed is specified by the user in terms of a set of parameters, including dataset and wavelength to be used, location and size on the sky, coordinate system and projection, and spatial sampling rate. Many astronomical datasets are massive, and are stored in distributed archives that are, in most cases, remote with respect to the available computational resources. Montage can be run on both single- and multi-processor computers, including clusters and grids. Standard grid tools are used to run Montage in the case where the data or computers used to construct a mosaic are located remotely on the Internet. This paper describes the architecture, algorithms, and usage of Montage as both a software toolkit and as a grid portal. Timing results are provided to show how Montage performance scales with number of processors on a cluster computer. In addition, we compare the performance of two methods of running Montage in parallel on a grid.