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The Pyrenean Haute Route - meaning 'High Route' (Haute Randonnée Pyrénéenne in French) is an 800+ km (about 500 miles) trail that follows the highest trekking route through the Pyrenees Mountains crossing the border multiple times between France, Spain and Andorra. The route starts from Hendaye in the East to Banyuls-sur-Mer in the West, although it can be walked in either direction. Unlike the waymaked trails of the GR10 (in France) and the GR11 (in Spain), the Haute Route is just that, a 'route' not a waymarked path; it does follow some sections so the GR10 & GR11 and other trails, but expect to use navigation and route finding skills for the numerous unmarked sections between. There are also numerous variations along the the Haute Route itself. The route described in the Cicerone Guide 'Trekking the Pyrenean Haute Route' is just the authors interpretation of the best/most scenic route. There is no right or wrong route.
The trekking season in the Pyrenees is from June to September with July and August being the high season, although August can be very hot and prone to thunderstorms. There can still be snow on the passes in the central region in June, and bad weather is common at the end of August and beginning of September. It will be getting colder as September goes on and there could be snowfall by the end of September.
There is an extensive network of mountains huts across the National Parks and the Pyrenees in general. Some may fill up quickly in the high season and need reservations. Camping is not technically permitted in the National parks, but there are camping areas by the mountain huts where overnight camping is permitted, and as long as small tents are sent up late and removed in the morning (8pm to 8am) then overnight stays are tolerated in the National Parks.
Have a look at the overview map that shows all the overnight points from the Cicerone guide: 'Trekking the Pyrenean Haute Route'. Also the maps you can buy and download to help you navigate along the trail.
The NEW Cicerone guide (2019) Pyrenean Haute Route