If you look closely at the curve of current as a function of voltage for a PN diode, it goes through zero (the origin, I = 0 and V = 0) with a finite slope (dI/dV).
To the left (reverse-bias) and right (forward-bias) of the origin, the shape of the curve is greatly different.
Now, if you re-graph the same data for forward bias on a semi-log scale (log I vs linear V), where I = 0 cannot be shown), you see that there is no actual threshold for conduction, it is just that the current increases approximately exponentially with the voltage for all positive voltage values.
The common approximations of 0.6 or 0.7 V as a threshold are just the voltages for a given current (e.g., 1 mA), where the voltage for a larger current (e.g., 10 mA) is not very different.
At zero and reverse-bias, the current is low and the capacitance is important: its variation with reverse bias can be exploited in varicap tuning diodes.