Poppies are perennials and will die back naturally and flower again next year. If you have ripe seed heads on them you could collect the seeds ( shake them out like pepper) and scatter them or keep them in an envelope and sow them in spring. Lots of poppies seed themselves randomly around the garden, anyway.
(You do not say which variety of poppy you have and that may be important in relation to the fertility of the seed.)

There are also the annual 'Shirley' poppies (Papaver rhoeas) which look similar to the poppy day appeal poppies. These grow and flower in the year and will automatically spread their seed and then die.

I seem to remember that most perennial poppies propagate by root division. Their seeds are usually non-viable. Annuals are notoriously good at self seeding. I have several varieties of each (I think there are dozens) and just leave them to their own devices apart from cutting back the annuals, for tidiness, once the seed pods are empty and, if I want extra perennials, I just divide the roots after they have flowered. Don't know if that is the 'correct' time to do it, but it works for me.

Perrenial poppies do propagate by root division - I agree and do the same Sarraceniac, but aren't they a nuisance for getting every bit of tuber out? The main root/tuber seems to go down for ever, and after splitting and transplanting I always spend some time digging up odd sprouts that appear just where I don't want them!!

Well if you leave bits of root then of course the poppy will continue to appear Avkq! However, most of us just leave our perennial poppies to get on and grown. Mec. Cambrica is almost impossible to get rid of and could be considered a weed!

I would never have thought of that, MG - inadvertently leaving bits of difficult to reach tubers and not expecting sprouts ... Shocking!

Ah well there you go... as I say it is almost impossible to remove Mec. cambrica roots completely - especially as they like to get into the roots of hedges and other shrubs...

That of course is only if you actually want to remove the plant completely. It's never bothered me for two reasons, one, when I put a perennial in I expect it to re-appear in more or less the same place every year for years and if I get a few extras - no prob, and, two, I'd rather have this 'problem' than nothing appearing at all. Lol.

Mec. cambrica seeds like the devil and even if we manage to dead head all our plants it comes in from neighbour's gardens hence to us it is a pretty but invasive weed.