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Nouakchott is the capital of Mauritania. Also it is the country's transportation hub - the main roads between Rosso, Nouadhibou, Atar and Nema all converge here. There are way to many honking taxis and their drivers still offering rides. It can get really annoying after some time. But stay open-minded, there are some nice people who really want to help.
North towards Nouadhibou, Morocco
Get to the police checkpoint a few kilometers outside of town - you can even hitchhike there as there is a lot of lorries going to just after the checkpoint. Zenit was treated very well by the police there in March 2011, even though they told him he couldn't hitchhike there and they would find a lift for him (which they did).
North towards Atar
Get to the end of town near the airstrip and walk a little bit until you come to a place where there's not so many houses near the road; from here you should be able to hitchhike out to police checkpoint.
East towards Kiffa, Ayoun el-Atrous, Nema, Mali
Again the best is to get behind the city to the police checkpoint.
South towards Rosso, Senegal
The best spot close to the city center is probably the Carrefour Madrid, where you could find a truck to at least the first police checkpoint out of town, but there is a lot of local traffic, most of which are taxis, and it is generally very crowded, so you might find it hard to attract the attention of drivers amongst the masses.
A shared taxi from the center to the outskirts is about 300UM per person.
Nouakchott is not a good place to be stranded in at night if you don't have money to pay for a camping spot in any of the auberges or manage to convince them to let you camp there for free... the church used to let people camp in their patio but don't anymore, and the beach is not very safe. The best strategy would probably be to ask at one of the police checkpoints just out of town if you might be allowed to crash out there for just one night.
Nouakchott also has a healthy couchsurfing community who all seem to know each other and it's quite easy to find a host there.