When my baby girl was born they told us to keep hats on her for sleeping for the first month at least and most of the hats we already had were a little big.  They would come down over her face as she squirmed around, which was very worrisome.  These hats on the other hand were a perfect fit for 0-2 months old and worked perfectly.  They are also very cute and hold up well to frequent washing.  They are also very soft.  I would recommend this hat set and may even purchase more.