There are several border points between Mauritania and Mali, but few offer very elegant ways of travelling. The most popular point for crossing into Mali, or the other way, are the roads between Ayunu l-Atrus in Mauritania and Nioro in Mali. Controls at the border points are unproblematic enough, as long as you have the necessary papers to enter the other country, and your belongings do not include anything illegal (like alcohol into Mauritania).
Unfortunately the border zone, is not the safest, and the remoteness of this region has invited to banditry. Care is therefore needed, and travelling with others will of course add to safety.
Nouakchott, being the most important point of arrivals into Mauritania, is actually not very well served by international flights. And beyond flying to Nouakchott, this capital has no other ways of connecting to the rest of the world.
Nouakchott has regular flights on Paris, France; Las Palmas, Spain; Dakar, Senegal; Jedda, Saudi Arabia; Algiers, Algeria; Tunis, Tunisia; Casablanca, Morocco; Banjul, Gambia; Conakry, Guinea; Niamey, Niger; N'djamena, Chad. Few of these operate more than 1 or 2 times per week, the services between Paris and Nouakchott, run 4 times per week.
Flying to Nouakchott from abroad is in general expensive, but there is room for good discounts for special types of tickets, particularly for students.
Nouadhibou serves as point of arrival for both overland travellers from Morocco, and for those travellers coming from the Canary Islands, either with a regular airliner, or with a cargo vessle or a fisher vessel.
Coming from Morocco into Mauritania, is easy enough, while going back the same way is illegal, and will need careful planning. Should you be cought by Mauritanian guards, they will not be cruel to anything else than your budget, as fines or bribing is made necessary.
Coming down the southern part of Morocco, will involve good planning, you own vehicle, as well as joining up with one of the two weekly convoys.
Las Palmas is a good point for arriving in Mauritania, and actually the cheapest alternative for most travellers coming from Europe or the Americas. Flying is apparently no longer done by Spanish Iberia, but Air Mauritanie should still be operating this flight. This flight will normally cost you about US$200-350. Getting across on a cargo vessel, will normally cost below US$30, and is quite a pleasant way of travelling.
Today crossing the border between Mauritania and Senegal is the easiest of all matters. While Rosso is the most important, there are several other crossing points, all across the Senegal river, and none with bridge, but with simple ferries: N'Diago, Diama, Jardu l-Moghayn, Takan, Laksayba, Boghe, M'Bagne, Kaédi, Tifunde Cive, Maghama, and Goraye.
The only rule is as simple: Have all papers necessary for entering the new country, as little of processing will be done at the border points.
At the present time, even getting near Tindouf, which lies 50 km north of the actual border between Mauritania and Algeria, is out of the question. Morocco, along with these two countries have made a zone with a variable radius of 200 to 600 km, inaccessible to foreigners, and heavily restricted even for nationals.
The reason for this situation is political, and connected to the politically unresolved conflict of Western Sahara, that was occupied by Morocco (and Mauritania, who left their part in 1979) in 1975. The army of Western Sahara, called Polisario, was for many years backed by Algeria, and have had their base in Tindouf. While there are no more fightings, the three countries are still far from agreeing on regulations on Western Sahara, and there will probably be years before travelling between Mauritania and Algeria can be done overland.