We explore some general cosmological aspects of brane localized gravity. In particular we show how brane embeddings can naturally lead to negative pressure quintessence models. We explore the embeddings of branes with non-zero spatial 3-curvature into a non-compactified AdS5 bulk, to find that for such branes it is not in fact possible to obtain an exponentially suppressed geometry away from the brane. Without needing to appeal to fine-tuning, we show how to naturally generate the thin Randall-Sundrum Minkowski brane starting from a thick brane generated by a sine-Gordon soliton. Using a "shine-Gordon" equation we show how to naturally self-compactify an AdS5 bulk into which a de Sitter brane has been embedded to thereby localize the geometry around it.