What happens one the breakdown voltage has been reached does the current conducted through a zener diode? I was thinking current increases drastically

 

Limited by any series resistance.
Max.

 

The maximum allowable current in amps = zener wattage/zener voltage.
For a 1 watt 12 volt zener the max current = 1w/12v ~ .083 amps.

Even though the rating is 1 watt, in real circuits you never run components at the max
wattage rating. My opinion is to be conservative, not more than about 1/4 rated wattage.

 

The maximum allowable current in amps = zener wattage/zener voltage.
For a 1 watt 12 volt zener the max current = 1w/12v ~ .083 amps.

OP, note that TubeGuy said, "Allowable", not "possible". Allowable means " the maximum cuerent that the designer should allow" and he/she should pick the right resistance to prevent higher currents from ever flowing through the Zener.

 

It will help to think about the i-v curve of a zener diode - as shown here:
http://www.learningaboutelectronics.com/Articles/Zener-diode-IV-characteristics-curve

Note that EVERY diode is a zener - just most 'general purpose' diodes have a very high breakdown/zener voltage (over 100V).

 

Zener diodes are connected in reverse bias for normal voltage regulation. When the voltage is below the rated voltage, the zener will not conduct. Anything above the rated voltage will be shunted to negative through the zener diode. However, a series resistor is required to limit current through the diode (almost the same as how an LED needs a current limiting resistor)

 

What happens one the breakdown voltage has been reached does the current conducted through a zener diode? I was thinking current increases drastically

Some zener datasheets quote the "dynamic resistance" - AFAIK; you treat the zener voltage as an absolute voltage (theoretical ideal zener) the dynamic resistance determines the VI curve after subtracting Vz.