An overview of research conducted by Key and colleagues (2006) suggests that vegetarians are quite healthy. Some of the research even suggests that they may be slightly healthier than non-vegetarians, although in this case, it's probably difficult to figure out what the proper comparison group would be since most people who are vegetarians by choice tend to be very health-conscious. Vegetarians seem to be less susceptible to the major killers (especially cardiovascular problems but also cancer).
There's very good evidence that vegetarian diets (particularly vegan diets) greatly reduced the odds of developing diabetes. The following was reported by Tonstad and colleagues (2011): Cases of diabetes developed in 0.54% of vegans, 1.08% of lacto ovo vegetarians, 1.29% of pesco vegetarians, 0.92% of semi-vegetarians and 2.12% of non-vegetarians. In multiple logistic regression analysis controlling for age, gender, education, income, television watching, physical activity, sleep, alcohol use, smoking and BMI, vegans (OR 0.381; 95% CI 0.236–0.617), lacto ovo vegetarians (OR 0.618; 95% CI 0.503–0.760) and semi-vegetarians (OR 0.486, 95% CI 0.312–0.755) had a lower risk of diabetes than non-vegetarians.
A study by Pettersen and others (2012) has shown that vegetarians (and especially vegans) have lower blood pressure--partly due to their lower weight.
Vegetarianism also seems to promote longevity. As Sabate points out, it isn't clear whether the benefits relate to the inclusion of so many healthy foods or the exclusion of meat.
A review by Bonnie and others (2011) shows that vegetarians tend to consume more essential nutrients.
Some research has indicated that vegetarians have lower colon and prostate cancer risks. (There's also an indication that legume consumption reduces the cancer risk created by eating red meat). It isn't clear whether the advantages of vegetarian diets are due to the increase in veggies or the omission of meat (for the former interpretation, see Hill, 2002).