Current US EPA Tier 2 Bin 5 (California ULEV)  NOx limits are max. 0.07 grams per mile. 
Current European (Euro 6) standards appear much easier to pass with max. 0.29 grams per mile.
Reporters have been saying 3rd party tests for VW's 2.0 TDI diesel NOx produced emissions levels between 10 to 40 times higher than US standards allow. Doing the math means the results ranged from 0.7 grams per mile to 2.8 grams per mile.
It is not easy to convert between Parts per million (PPM) to Grams per Mile (g./mi.), since PPM is an instantaneous measurement ratio , whereas g/mi. is a volume measure over distance. To convert, you would need to know other factors such as vehicle speed, fuel consumption, etc. for a particular distance travelled. 
I think emissions testing stations call this bag testing. IE. collect all tailpipe emissions over a driven distance and then analyze mass of the various emission components. (particulates, CO, HC, NOX, etc.)
So take the chart below with a grain of salt, as I am not a scientist. 

  Using a government issued chart from the 1970's and average vehicle parameters, a ballpark relationship between PPM and grams/mile is illustrated.
The chart intersection of the current max NOx limit of 0.07 grams/mile is at about 0.02 PPM. 
<edit>I think this chart is suggesting that current automobile NOx limits are now much lower than the ambient levels of NOx were in the 70's. L.A. smog days were off the chart.<edit>

Many of us who have replaced our 15-26 year old Metro/Swift catalytic convertor, and then immediately go for tailpipe emission testing (Dynamometer-road test, not OBD test) have seen results in the (
Corrected:  0 to 10 PPM range for NOx, or 0.03 g/mi during driving test)