The short answer is Yes, oats are gluten-free however it's not that straightforward.
Growing up I was always told that Oats were a caution food for Coeliacs, some coeliacs could tolerate them whereas others couldn't.
Oats don't actually contain gluten, they contain something called "avenin" which is very similar to gluten. However, according to the Coeliac UK website, research has shown that most coeliacs can safely eat avenin.
So what's the problem with Oats?
The main problem with oats is when they're produced in the same environment as non-gluten-free products such as: Wheat/Barley/Rye, all of which contain gluten. Cross-contamination can occur which results in the oats not being gluten-free and as such, not suitable for Coeliacs.
Luckily, there has been a recent rise in gluten-free oat products suitable for Coeliacs; such as these delicious gluten-free oat crackers from Nairn's. Products like these are produced in a gluten-free environment and are clearly labelled with the crossed grain coeliac symbol.
All the research points to oats being naturally gluten-free, but be sure to check for specific gluten-free oats to avoid any cross contamination. If you're unsure about trying oats for the first time, check with your doctor first!