This communication presents the situation regarding astronomical and archaeoastronomical heritage related to the World Heritage Convention through recent years up until today. Some parallel events and works were promoted strongly within the IAU–UNESCO Initiative during the International Year of Astronomy (2009). This was followed by a joint program by the IAU and ICOMOS—an official advisory body assisting the World Heritage Committee in the evaluation of nomination dossiers. The result of that work is an important publication by around 40 authors from 20 different countries all around the world: Heritage Sites of Astronomy and Archaeoastronomy in the Context of the UNESCO World Heritage Convention (Ruggles & Cotte 2010). A second volume is under preparation (2015). It was also accompanied by some initiatives such as the “Windows to the Universe" organisation and the parallel constitution of local “Starlight Reserves”. Some regional meetings studying specific facets or regional heritage in the field giving significant knowledge progresses also accompanied the global trend for astronomical heritage. WH assessment is defined by a relatively strict format and methodology. A key phrase is “demonstration of Outstanding Universal Value” to justify the WH Listing by the Committee. This communication first examines the requirements and evaluation practices about of demonstrating OUV for a given place in the context of astronomical or archaeoastronomical heritage. That means the examination of the tangible attributes, an inventory of the property in terms of immoveable and moveable components and an inventory of intangible issues related to the history (history of the place in the context of the history of astronomy and cultural history). This is also related to the application to the site of the concept of integrity and authenticity, as regards the place itself and in comparison with other similar places (WH sites already listed, sites on national WH Tentative Lists, or other similar places in the region). The second issue of the communication is to give a glimpse of today's WH List, including some difficulties with listing and occasional failures, and trends and promising approaches.