Carbon nanohorns (sometimes also known as nanocones) are conical carbon nanostructures constructed from an sp(2) carbon sheet. Nanohorns require no metal catalyst in their synthesis, and can be produced in industrial quantities. They provide a realistic and useful alternative to carbon nanotubes, and possibly graphene, in a wide range of applications. They also have their own unique behavior due to their specific conical morphology. However, their research and development has been slowed by several factors, notably during synthesis, they aggregate into spherical clusters similar to 100 nm in diameter, blocking functionalization and treatment of individual nanocones. This limitation has recently been overcome with a new approach to separating these "dahlia-like" clusters into individual nanocones. In this review, we describe the structure, synthesis, and topology of carbon nanohorns, and provide a detailed review of nanohorn chemistry.