The current chapter divisions were a scheme devised by Stephen Langton, sometime around 1204-5. The claim is often made, but for a good analysis of the sources and documentation to support the claim, I would read Roger Pearse's treatment online here. This was not the only attempt to divide the Bible into chapters, but it was the one that was successful enough to became the standard.

Modern verse divisions were not added until Robert Stephanus (aka Robert Estienne), printer of a Greek-Latin New Testament in 1551. OT verse references to some extent, though not completely, follow division markers in the Hebrew text. Those markers (sentence and paragraph markers for the most part) probably date back to the work of the Masoretes, but didn't become widely used as a numbering system until the 15th century. In 1661 a numbered Hebrew Bible was printed by Joseph Athias of Amsterdam.