Using one material system from the near infrared into the ultraviolet is an attractive goal, and may be achieved with (In,Al,Ga)N. This III-N material system, famous for enabling blue and white solid-state lighting, has been pushing towards longer wavelengths in more recent years. With a bandgap of about 0.7 eV, InN can emit light in the near infrared, potentially overlapping with the part of the electromagnetic spectrum currently dominated by III-As and III-P technology. As has been the case in these other III-V material systems, nanostructures such as quantum dots and quantum dashes provide additional benefits towards optoelectronic devices. In the case of InN, these nanostructures have been in the development stage for some time, with more recent developments allowing for InN quantum dots and dashes to be incorporated into larger device structures. This review will detail the current state of metalorganic chemical vapor deposition of InN nanostructures, focusing on how precursor choices, crystallographic orientation, and other growth parameters affect the deposition. The optical properties of InN nanostructures will also be assessed, with an eye towards the fabrication of optoelectronic devices such as light-emitting diodes, laser diodes, and photodetectors.