In a nutshell (no pun intended), the vegan diet can contain all the nutrients your body needs, but you need to plan very carefully, have as wide a range of vegan foods as you can (including more unusual foods!) and be prepared to take the odd supplement and some fortified foods.
Those eating a plant-based diet should focus on vegetables, fruit, legumes and whole grains and add some nuts and seeds and take a good vitamin B12 supplement daily. Also consider taking an omega 3, vitamin D and iodine supplement as a safeguard.
Provided the above considerations are made, a vegan diet can be very healthy; often providing more antioxidants and phytochemicals, potassium and magnesium. It also tends to be lower in fat, and higher in fibre, thus resulting in a healthy gut.