To and from EGYPT
There are fair airline connections between Cairo and Khartoum, and prices are normally around US$250 one way. Connections overland would mean taking a boat from Aswan to Wadi Halfa, and this will cost US$20-30 one way. Driving this way (from Abu Simbel and down Lake Nasser, on the other side from Wadi Halfa) seems to be prohibited. Driving along the Red Sea Coast is also prohibited, due to political disagreements on the territories out here (the Sudan is controlling land that is Egyptian). The last possibility is travelling with caravans. This is easiest done if you catch up with these in the Sudan, in the Darfur area.
To and from LIBYA
There are lorries doing the trip between the two countries, but this takes time. And try to make sure your papers are OK, before setting off.
To and from CHAD
There are overland connections connecting Al Junayna to the unstable Chad. This will most likely be lorries, and getting on these shouldn't be all too difficult. But the area you will be travelling through, is unsafe from bandits.
To and from CENTRAL AFRICAN REPUBLIC
The same applies to Central African Republic as to Chad. Unrest in the country at the present stage should make you feel alert though.
To and from CONGO
Border crossings are made in the same manner as with Chad and Central African Republic. If you make it to the border, the civil war in southern Sudan will make this journey not advisable.
To and from UGANDA
Somewhat better connections between the Sudan and Uganda, with buses doing the 200 km between Juba and the border posts, in addition to the lorries. But as for Zaïre, civil war in southern Sudan, as well as unrest in northern Uganda are factors you have to consider seriously.
To and from KENYA
Much the same as for Uganda, but roads are far worse. Many people use the Ugandan connections when coming from Kenya to the Sudan, or vice versa.
To and from ETHIOPIA
Apparently no connections. You will have to go through Eritrea.
To and from ERITREA
There are good bus connections between Asmara, the capital of Eritrea, and Port Sudan and Khartoum.
There are some lines connecting European cities directly to the Sudan. Count on using at least €800, or US$1,000 for a round trip.