Can I put raised beds onto concrete?

Yes, J&J, I don't see why you should not have a raised bed on a concrete slab. You will need to make sure that water can easily escape from below the wall of the bed onto the concrete. Is the concrete very thick or can you 'punch' a few holes in it for drainage? The walls of this bed will have to be fairly deep as your plants won't be able to push their roots down into the soil beneath the bed. I would try to put some sort of barrier between the raised bed and the fence to keep the damp soil away from the fence and spoiling it.
We have a similar concrete shed base that we have been using as our patio for years. We have troughs and planters on it along with our garden furniture and it works a treat.

Agree with bulbaholic - if you are building a raised bed with brick walls, you will need bricks at the back too - the fence will rot very quickly if soil is in contact with it. You will also need to enable drainage from the soil you put in the bed, either by punching holes in the concrete base, or leaving drainage channels at the bottom of the brickwork wall. It will need to be a minimum of 2 feet deep to grow anything much, and that precludes growing anything shrub or tree like that reaches more than about a foot or so in height. You will also need to leave a minimum clearance between the fence and the back of the raised bed of six inches, preferably more, to prevent damp building up in the wood of the fence causing it to rot more quickly.

Fantastic, thank you so much for those prompt replies. Really helpful!

Was thinking of growing bamboo or a similar grass in the back of the beds to provide a screen from next door - is this possible or is there not going to be enough depth for the roots?

The roots of bamboo and grass are likely to be shallow but will spread widely; this is probably not good in the confined space of a raised bed. You could try these items in containers plunged in the bed to contain the roots.

I was going to say bamboo should be okay in 2 feet of soil, but you won't be growing anything else in there with it!

And who should know better than Bamboo :-)

:) thank you.