Research shows a decreased risk of SIDS when babies are placed flat on their backs for sleeping (specifically, on a firm mattress with no loose bedding).
There is not a universal age for safe tummy sleep because all babies are different with skill development. However, once your baby can purposefully and consistently roll from back to tummy and tummy to back, most pediatricians will say tummy-sleeping is safe. (Ask your doctor though.)
Parents should still continue to place a baby on his back for sleeping, but if he is strong enough to roll onto his tummy, most doctors will agree he is strong enough to lift his head and safely sleep in this position. This same principle applies if your baby seems to be face down after rolling over onto his belly; if he has the strength to roll over, he has the strength to lift or turn his head if he needs to while on his belly (again, ask your pediatrician.)
However, babies MUST NOT be swaddled if sleeping on their tummies for safety reasons. I’ve got a blog just for you on transitioning out of the swaddle that can help. Also, the crib must be free of all loose bedding and bumpers. My safe sleep checklist covers all things safe sleep.
You may find that your baby is so accustomed to sleeping on his back that flipping to his tummy and getting stuck there turns him into one very angry little person. This is what happened when my son started rolling over onto his belly, and I remember flipping him over 1,143 times one night! That may be a bit of an exaggeration, but it sure felt like that!
You don't have to go in and roll your baby over onto his back each time he flips to his belly, but expect that it will take time for him to learn to like this new sleeping position.
During the day, practice tummy time frequently.
If he awakens in the night after flipping to his tummy, resist the urge to roll him to his back (or get ready to do it 1,142 more times). Instead, rub his back, pat his bottom, and "shhhh" him to help him learn to fall asleep while on his tummy.
Most likely, this will take about 4-7 nights.
There's one product I love for this: Merlin's Magic Sleep Suit prevents young babies from rolling while in the crib and helps maintain that favorite back-sleeping position. The sleep suit is a great tool and is safe to use until your baby can roll in the suit.
Rolling over and tummy sleep may begin when babies are 3-4 months old, a time filled with developmental advances and sleep challenges. If you are in this tricky phase, I have a resource for you! Navigating Months 3 & 4 is a 50-page ebook that will guide you through the dreaded 4-month sleep regression and help you on your journey towards more restful nights.
If your baby reaches 5 months and is still struggling, I have classes that can help. The 5-24 Month Collection will walk you step-by-step through a plan to conquer nights, days, and every bump along your sleep journey.