Through seed but more particularly by offset bulbs in the soil, at first tiny, then growing. Spread like wildfire, only solution is to carefully dig out, removing all bulbs and bulblets. Even so, you'll miss some.

..and deadhead them, just in case they are seeding.

Roundup does work on them, but you have to get it to stick to the leaves. Add a little liquid soap to the mix before spraying.

I did the same and now seem to be forever pulling them out. MIne are growing in amongst the roots of other plants and are a real pain. Just treat them like weeds and persevere with pulling them out every week. It will work ........eventually.

Only if you pull the bulbs out with the foliage, Sally...

I planted these last year, as pretty as they are after I heard about how they multiply I have dug them up - some new bulbs were as small as pearls, so may not have got them all.

I have had terrible trouble with these Alliums. Even yesterday before it rained I was planting the border and came across lots of minute bulbs in the soil in one spot. If these are the same as mine, they leaf up in the winter when other plants are dormant and bloom quite late in spring. That is when the Round Up can be used more easily. The snow stopped me doing this last winter. I had to completely clear the border.