In SPSS, is it possible to export a dataset file to .CSV with the value names instead of the value numbers?

The other answers relate to the GUI, but some people prefer using the syntax editor. If you fall into that category, from the programming perspective, you can do something like:

SAVE TRANSLATE OUTFILE='path\to\file.csv'
  /TYPE=CSV
  /MAP
  /REPLACE
  /FIELDNAMES
  /CELLS=LABELS.

Notice the last line, /CELLS=LABELS. -- If you wanted the values instead, you can change it to (surprise!) /CELLS=VALUES.

Yes, it is possible (at least in SPSS 20). In File Menu, choose, "Save As...", in the drop down menu in the dialog box that opens choose "Save as type: 'Comma delimited (*.csv)')", and underneath that select "Save value labels where defined instead of data values" (or hit "Alt-a") and choose "Save".

In my version of SPSS (17), the save window has a checkbox for "Save value labels where defined instead of data values", which will save the accompanying labels rather than numerical values.