Unfortunately there is not a straight forward answer to this question.
Oats are the seed of the cereal plant Avena sativa or A. byzantina. Although they do not contain gluten they do contain the protein avenin which is similar to gluten. Most people who react to gluten do not react to this protein but some people do.
However due to high possibility of cross contamination during planting, harvesting and processing many brands of oats do have detectable amount of gluten in them. Therefore to be safe and avoid any issues, oats are usually grouped with the grains containing gluten; however in reality oats may have a place in a gluten free diet. Oats are commonly included in breakfast foods and health products, as well as in some baked products.