The fibers used in artificial turf fields have taken huge strides in the past decade.  About ten years ago, new forms of artificial turf began replacing AstroTurf called FieldTurf.  This new form of turf was thicker, softer, and better replicated natural grass.  Today, many municipal, school, college, and professional sports fields use these new artificial turf surfaces as it less expensive and stays in better condition than a grass field under heavy use. 
A professional sport played on field turf

Seattle's Qwest Field With FieldTurf

Maintenance, trimming, chalking boundary or other field lines, and irrigation are not needed with an artificial turf surface, which results in large cost savings.  Artificial turf fields can also be used for a variety of sports such as football, soccer, softball, baseball, Frisbee football, field hockey, and lacrosse.  They are truly a multi-purpose surface that can be played on by different sports throughout the year. Grass fields are subject to rapid wear

with use, especially when conditions are wet.  Grass fields do not drain quickly and rainfall can quickly turn the field into a torn up mud pit with use.  A synthetic turf surface is designed to drain quickly and does not suffer wear excess wear with use.  There are no puddles, grass clumps, or mud.  A high quality, durable artificial surface can last for years, unlike natural grass, which can literally be torn apart after one game. 

Also, most players love playing on new artificial turf surfaces.  They are soft, forgiving and durable, not wearing down during play as a grass field often does.   There is no mud or puddles.  They cause much fewer injuries than Astroturf did, as they are softer and not as abrasive.  Artificial surfaces today may even be an improvement on nature’s creation of natural grass.