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 :)Hi, I'm very happy to have just gotten a small plot which came with an old apple tree. The branches though have this white fluff. Should I be worried? ??? Thanks in advance for any advice as I am clueless about apple trees. (I come from a tropical country! ;D)

is the white fluff under leaves or actualy on the branches??

sounds like wooly aphid.
Also called american blight if my memory serves me

Yes, it's on the branches. (Wish I had a digital camera!) Not powdery or moldy but fluffy.

Oh no, I finally found a picture of it and I think it is wooly aphid!

That would be my first guess too, David (assuming it is around leaves and buds rather than on old bark). If you tease apart the fluff you will eventually find aphids among it.
Based on an experience with them years ago at my Mum's place, my advice would be KILL THEM NOW!  >:(  They can spread like wildfire, promote fungal infection, and are hard to kill even if you try to squash them all by hand.

Our experience was that they were very difficult to get rid of using anything other than repeated treatment with fairly unpleasant systemics, probably because a) they're aphids and by the time you kill one it's probably had 100s/1000s of babies and b) the fluff protects them from any treatment which has to touch them (eg. soap) and hides them from predators.

 :o :o :o Guess I'll have to put my exterminator hat on!

Scrub with Methylated Spirits!

am 100% with Tim .... scrub with Meths

we had some last year and used and old toothbrush to apply the Meths, two treatments and most had gone. When Meths is applied the white fluff turns Pink (or ours did  ;D) which made it easy to see where we had been.

We only got it on one tree, the oldest we have (out of five) by a long way.

Mikey

 ;) Thanks for all the advice, happy gardening!