DNA has become one of the most extensively used molecular building blocks for engineering self-assembling materials. DNA origami is a technique that uses hundreds of short DNA oligonucleotides, called staple strands, to fold a long single-stranded DNA, which is called a scaffold strand, into various designer nanoscale architectures. DNA origami has dramatically improved the complexity and scalability of DNA nanostructures. Due to its high degree of customization and spatial addressability, DNA origami provides a versatile platform with which to engineer nanoscale structures and devices that can sense, compute, and actuate. These capabilities open up opportunities for a broad range of applications in chemistry, biology, physics, material science, and computer science that have often required programmed spatial control of molecules and atoms in three-dimensional (3D) space. This review provides a comprehensive survey of recent developments in DNA origami structure, design, assembly, and directed self-assembly, as well as its broad applications.