Intersecting brane geometries

We present a survey of the calibrated geometries arising in the study of the local singularity structure of supersymmetric fivebranes in M-theory. We pay particular attention to the geometries of 4-planes in eight dimensions, for which we present some new results as well as many details of the computations. We also analyse the possible generalised self-dualities which these geometries can afford.Comment: 29 pages. (v2: Two new sections have been added. Paper is now more than 50% longer. A guided tour of the eight-dimensional geometries is presented along with the computational details. We have also included a discussion of the possible generalised self-dualities afforded by these geometries.