I understand that the ZonalStatistics (ArcToolbox -> Spatial Analyst Tools->Zonal->Zonal Statistics) in ArcGIS (10.3) allows the extraction of raster values within zones defined by a polygon shapefile (takes the average over the specified areas). ZonalStatistics creates a new raster file based on the polygons. I'd like to directly add the raster values to my polygon shapefile as a new column.
If you look in the toolbox, exactly 1 tool below Zonal Statistics is a tool called Zonal Statistics as Table! This creates a table which you can then join back to your original polygon dataset.
Add Surface information will add values from a raster to the attribute table of a polygon feature class. This tool requires a 3D Analyst toolbox and an advanced license.