This paper describes version 2 of the OI Exchange Format (OIFITS), the standard for exchanging calibrated data from optical (visible/infrared) interferometers. This IAU-endorsed standard has been in use for 10 years at most of the past and current optical interferometer projects, including COAST, NPOI, IOTA, CHARA, VLTI, PTI and the Keck interferometer. Software is available for reading, writing and merging OI Exchange Format files. This version 2 provides definitions of additional data tables (e.g. for polarisation measurements), addressing the needs of future interferometric instruments. Also included are data columns for a more rigorous description of measurement errors and their correlations. In that, this document is a step towards the design of a common data model for optical interferometry. Finally, the main OIFITS header is expanded with several new keywords summarising the content to allow data base searches.