How do I export SVG images from multiple artboards in a batch process using Adobe Illustrator CC
If you're using CS6 (not sure about earlier versions) then you need to save as SVG, there is no SVG export option.
File → Save As → SVG (svg)
In CC there is a new export option for SVG.
File → Export → SVG (svg)
In both cases, make sure to check "Use Artboards" to export the contents of your artboards as individual SVG files.
File > Export > Export for screens.
Set the format to SVG.
This pretty powerful and allows you to select each artboard (rather than asset) you want to export. You can also export multiple formats in one go (great for backup PNGs for example).