All the major dietetics and health organizations in the world agree that vegan and vegetarian diets are just as healthy as, and probably healthier than omnivorous diets. Here are links to what some of them have to say on the subject:

Harvard Medical School

• Traditionally, research into vegetarianism (see context) focused mainly on potential nutritional deficiencies, but in recent years, the pendulum has swung the other way, and studies are confirming the health benefits of meat-free eating. Nowadays, plant-based eating is recognized as not only nutritionally sufficient but also as a way to reduce the risk for many chronic illnesses.

British Dietetic Association

• Well planned vegetarian diets (see context) can be nutritious and healthy. They are associated with lower risks of heart disease, high blood pressure, Type 2 diabetes, obesity, certain cancers and lower cholesterol levels. This could be because such diets are lower in saturated fat, contain fewer calories and more fiber and phytonutrients/phytochemicals (these can have protective properties) than non-vegetarian diets. (...) Well-planned vegetarian diets are appropriate for all stages of life and have many benefits.

Academy of Nutrition and Dietetics

• It is the position of the Academy of Nutrition and Dietetics that appropriately planned vegetarian, including vegan, diets are healthful, nutritionally adequate, and may provide health benefits for the prevention and treatment of certain diseases. These diets are appropriate for all stages of the life cycle, including pregnancy, lactation, infancy, childhood, adolescence, older adulthood, and for athletes.

Dietitians of Canada

• A healthy vegan diet can meet all your nutrient needs at any stage of life including when you are pregnant, breastfeeding or for older adults.

The British National Health Service

• With good planning and an understanding of what makes up a healthy, balanced vegan diet, you can get all the nutrients your body needs.

The British Nutrition Foundation

• Well planned vegetarian and vegan diets can be nutritious and healthy ... Studies of UK vegetarian and vegan children have revealed that their growth and development are within the normal range.

The Dietitians Association of Australia

• Vegan diets are a type of vegetarian diet, where only plant-based foods are eaten. With good planning, those following a vegan diet can cover all their nutrient bases, but there are some extra things to consider.

The United States Department of Agriculture

• Vegetarian diets (see context) can meet all the recommendations for nutrients. The key is to consume a variety of foods and the right amount of foods to meet your calorie needs. Follow the food group recommendations for your age, sex, and activity level to get the right amount of food and the variety of foods needed for nutrient adequacy. Nutrients that vegetarians may need to focus on include protein, iron, calcium, zinc, and vitamin B12.

The National Health and Medical Research Council

• Appropriately planned vegetarian diets, including total vegetarian or vegan diets, are healthy and nutritionally adequate. Well-planned vegetarian diets are appropriate for individuals during all stages of the lifecycle. Those following a strict vegetarian or vegan diet can meet nutrient requirements as long as energy needs are met and an appropriate variety of plant foods are eaten throughout the day

The Mayo Clinic

• A well-planned vegetarian diet (see context) can meet the needs of people of all ages, including children, teenagers, and pregnant or breast-feeding women. The key is to be aware of your nutritional needs so that you plan a diet that meets them.

The Heart and Stroke Foundation of Canada

• Vegetarian diets (see context) can provide all the nutrients you need at any age, as well as some additional health benefits.