Yes you grow your eyebrows without surgery. There are special home treatments such as Latissee or Revitalash that is used not only to help eyelashes grow longer and thicker but it can also be used to help eyebrows to grow thicker and darker as well. You need to be dedicated when using the treatment. It needs to be applied every night before bed for at least 4 months. This is the a very alternative to surgery. With patience and dedication, you can begin to see good results in about 2 months. Good luck!!!
There are two non surgical options - minoxidil (Rogaine) and bimatoprost (latisse). These are topical options and can help a proportion of individuals improve their eyebrow density.