Multivariate data is often visualized using linear projections, produced by techniques such as principal component analysis, linear discriminant analysis, and projection pursuit. A problem with projections is that they obscure low and high density regions near the center of the distribution. Sections, or slices, can help to reveal them. This paper develops a section pursuit method, building on the extensive work in projection pursuit, to search for interesting slices of the data. Linear projections are used to define sections of the parameter space, and to calculate interestingness by comparing the distribution of observations, inside and outside a section. By optimizing this index, it is possible to reveal features such as holes (low density) or grains (high density). The optimization is incorporated into a guided tour so that the search for structure can be dynamic. The approach can be useful for problems when data distributions depart from uniform or normal, as in visually exploring nonlinear manifolds, and functions in multivariate space. Two applications of section pursuit are shown: exploring decision boundaries from classification models, and exploring subspaces induced by complex inequality conditions from multiple parameter model. The new methods are available in R, in the tourr package.