BAA consists of a number of formal and informal bodies that govern its activities.

The supreme body of the organisation is the General Conference of all its members that meets at least once every two years. In the period between the sessions of the General Conference BAA's leading body is the Board of Directors, an eleven-member body elected by the General Conference. The BAA Board makes all the important decisions concerning the organisation, including the appointment of the Executive Director. The latter appoints his Staff.

Along with the Board of Directors, there is the three-member BAA Internal Auditing Committee (IAC), which is appointed by the General Conference. The IAC is in charge of the internal financial supervision of the projects implemented by BAA.

Apart from the formal bodies, there are also some informal clubs within BAA. The most prominent among these clubs are the Armenian John Smith Fellows' Club (JSFC) consisting of those BAA members that have accomplished the annual John Smith Fellowship Programme in the UK, the Eurasia Centre Scholars, consisting of those scientists and researchers who have taken part in the Eurasia Centre's research program at the University of Cambridge, the Oxford Alumni, LSE Alumni and Essex Alumni Clubs and several others.

The first Honorary President of the BAA was the former British Ambassador to Armenia Mr. Timothy Jones, who was succeded by HBM Ambassador Ms. Thorda Abbott-Watt. The current Honorary President is HBM Ambassador to Armenia Mr. Anthony Cantor.