How to measure the inverse saturation current of a diode?
The last time I made some diode leakage measurements, I used a cheap (distributor's own brand, &lt;£10) DMM, with 1999 full count, 10M input impedance, and a 200mV range, as the current meter. 

On that range, the full scale is 200mV/10M = 20nA, with the nominal current resolution 100uV/10M = 10pA. 

You can use an external shunt to get a higher current range for big diodes, use an external 1.1M resistor for 200nA full scale, and 100k for 2uA.

It's worth using a variable power supply, and inching it up from zero, so as not to embarrass your meter's 200mV range with a leaky or failed diode. I found (and the Schockly equation will tell you) that once above a couple of volts, the reverse leakage current is more or less constant.

FWIW, I measured about 35nA for a BAT42 (schottky), 4nA for a 1N4148, and failed to measure (so &lt;10pA) the current for a BAS116 which is advertised as a 'low leakage' diode.

If you want to measure lower currents, then you need to build a pico-ammeter round a low bias op-amp, next-hack's answer shows you how.

Two cautions when working like this. (1) You might want to verify that your meter is actually 10M input impedance on the 200mV range, put an external 10M in series with it, and check that halves the reading when supplied with a voltage. (2) When using non-native ranges like this, the decimal point will usually be in the wrong place. I am skilled at moving it the wrong way in my head, so I simply enter the voltage I read into a spreadsheet, and the relevant resistance in another column, and let it do the sums.