John Roe [15] introduced the notion of coarse cohomology of a metric space to studylarge scale geometry of the space. Coarse cohomology of a metric space roughly measures the way in which uniformly large bounded set in that space fit together. In the first part of this dissertation, we describe a joint work with Boris Okun that generalizes Roe’s theory to define coarse (co)homology of complement of any given subspace in a metric space. Inspired by the work of Kapovich and Kleiner [12], we introduce a notion of a manifold like object in the coarse category (called coarse PD(n) space) and prove a coarse version of the Alexander duality for these spaces. In the second part of this dissertation, we generalize a Theorem of Roe [15] to compute coarse cohomology of the complement for many spaces by relating coarse cohomology of the complement with the Alexander–Spanier cohomology. In the final part of this dissertation, we introduce an equivariant version of coarse cohomology of the complement. We then use this theory to find obstruction to coarse embedding of a given space into any uniformly contractible n-manifold. i
