The LB7 Duramax from 2001 to 2004.5 had injector problems. GM ended up offering an extended warranty on the injectors. I think it was around 200,000 miles or 10 years. To make matters worse the injectors are hard to get to and require partially disassembling the engine.
The LLY Duramax from 2004.5 to 2005 had relatively few problems. Some claim they overheat but I have one with 205,000 miles on it that does not so I don't know how much to believe that claim.
The LBZ Duramax from 2006 to 2007.5 came with the 6 speed Allison instead of the 5 speed. It also had increased HP and no DPF or DEF. These are considered the best of the best Duramax. If you find one you better buy it because it won't be for sale for long.
The 2007.5 to 2010 LMM was basically reliable and trouble free. It gained the DPF. The 2011 - 2015 Duramax LML has also been pretty reliable and trouble free. The Bosch CP4 injection pump has given some issues but GM has tended to fix them under warranty. The same pump is used on the 6.7 Ford with similar issues but Ford seems to blame the problem on the customer and not repair under warranty.