Oh, such a familiar story - the patio is covered in weeds, there are terrible things growing up between the slabs, it looks awful, it's horrible to walk across as the willowherb and grass tickle your ankles, and what is to be done?
Most people turn to weedkiller as being the easy answer, but when your weeds have reached the ankle-tickling stage, it is the wrong thing to do.
Because although the weedkiller will kill off the plants, that is not the whole story: the dead plant matter will fall back to the gaps in the slabs, adding to the rich organic mix down there, and providing the perfect germination and growing medium for yet more weeds. No to mention having to look at the brown and dying corpses for weeks or months.
Long-term weedkillers such as Pathclear are designed to kill off top-growth and to prevent new seeds germinating: but if you use it on "big" weeds, only the outer shell of the plant is coated with the chemicals, meaning that all the soft inner tissues are "clean" and, as plants generally rot from the outside first, these clean inner tissues form a good organic layer on top of the "you can't germinate here" layer. Perfect for new weeds.
And if you use glyphosate-based (translocated) weedkillers, well, one of their biggest selling points is that they are inactivated on contact with the soil, so once they have killed the weeds, the remaining plant matter is almost by definition a good seedbed.
So what is the right answer? One of the those flame-wands, with a gas canister and a long handle? I should say not, despite them being marketed as being eco-friendly: I just looked at one that proudly stated "Environmentally friendly; no hazardous chemicals". No nasty chemicals? How can any grown adult not see that a flame wand is made of a pressurised aerosol can, fuel (butane), metal tubing, ignition devices, safety devices, plastic handle etc etc - how can you not see just how non-eco these items are to make? How many chemicals, processes, how much energy was used in producing these things? And it's not even the one-off purchase of the tool - you have to keep buying new canisters of fuel for them, and in my experience they are not particularly effective, either: they may well scorch off the top growth, which is slow but satisfying to do, but the weeds quickly grow back as their roots are inevitable undamaged, so most of these toys end up collecting dust and webs in the shed, which is so incredibly wasteful.
No, the only real answer is to get out there and pull the weeds out by hand. Not what you were wanting to hear, but truly this is the only way to do it properly.
Put on your knee pads, get yourself a narrow tool - the good old Daisy Grubber being my first, last and best choice - get a bucket for the bits, and start weeding.
Start in one place, at random: take hold of the top growth in one hand, and poke the Daisy Grubber down between the slabs. Wiggle it around until you feel the plant start to move, then pull gently and steadily upwards to ease out the entire plant, including roots. If you just pull them, you will snap off the top growth but leave the roots undamaged, meaning that they will grow back even bigger than before, so it is very much worth the effort to get them out undamaged. And it's actually quite satisfying, when you succeed in getting a great long length of root out in one piece!
Move your way along the crack, clearing every weed as you go. When you reach a join, it doesn't matter which way you go, as long as you are steadily clearing out all the green matter. Put all the debris into the bucket or trug as you go, to reduce the amount of clearing up to do afterwards, and to avoid accidentally pushing dislodged weeds back down into the cleared channels.
The earlier groaning, by the way, about Alchemilla is due to the very strong, solid roots that this plant creates: it's a horrible job to remove it from cracks in paving and in steps, and over the years I have found that the trick is a) to catch them while they are tiny plants and easy to remove, and b) - yes, I said trick, but actually there are two parts to this trick - to use the Daisy Grubber against the actual slab to lever them up. Stab the forks down into the roots of the Alchemilla, as far "underground" as you can, then lever it out. If it snaps, repeat the process, stabbing and wiggling and levering until you can get the whole clump out. If, however, you've inadvertently left it too late, and they are solid lumps of root, the only answer will be to apply glyphosate -based weedkiller, wait until they go brown and die, then lever out the remains, which will hopefully shrink away from the sides of the slabs as they die.
If you have loose pea shingle between the slabs, this is a bit of a pain: shake off as much of it as you can from the roots, while trying not to get too much soil back amongst it: then when you have finished the weeding, scrape it back into the channels, then sweep any last bits away. If you only have earth between the slabs, dig out the earth as well, to a depth of about an inch, in order to get out all the scraps of root, and all the waiting seeds.
Once you have done all this, you should have a clear patio with clear channels round each slab. Spray carefully along each channel, or each crack/join if you don't have channels, with Pathclear. This is the weedkiller which prevents germination.
Then fill the channels with either sand, or a fresh layer of loose pea shingle. Spray the channels again with Pathclear.
This should allow you a good 6 months of weed-free patio - a whole year, if you are lucky. Just keep an eye out for new growth, and keep a small spray-bottle of glyphosate handy, to spritz any green bits as soon as you see them.
Then, hopefully, you won't have to spend another afternoon bend double!