Condoms do have an expiry date and you can see that mentioned in every pack. But, it is generally inferred that condoms can last upto 3 years after their expiry date and it varies between brands. The older they get beyond their expiry date; those condoms lose their effectiveness in preventing pregnancy or STDs.
One way to know your condom expiry date is to look at the condom pack and the other way is to open the pack and see if the condom is too dry or too sticky.
In some cases, you can find condoms too stiff and are not flexible enough. It is a clear indication that condom is at the fag end of its effective life and you better buy a fresh condom to be sure of its effectiveness.