Why are there no longer any biplanes?
Firstly, there are:
For example this company currently makes a biplane
But the main reason is that they're inefficient, and don't really do much that a monoplane can't.
They produce a lot of drag compared to the amount of lift they produce, meaning they use a lot of fuel to travel the same distance. They have fairly poor visibility, which is usually considered a bad thing!
They do have a few advantages - they have a good roll rate, making them quite maneuverable, and they have a very low stall speed and can fly on little power, meaning they can fly slowly very well (where their increased drag doesn't make much difference: drag increases with the square of airspeed)
Overall, though, the above makes them suited only to aerobatics, which can also be performed by monoplanes. Advances in wing design and composite materials take away most of the advantages of a biplane for any other use: ie actually travelling (the main purpose of an aeroplane) and carrying a load.
Overall, then, they're more complex than a monoplane, less efficient, and aren't much more manoeuverable. They have more disadvantages than advantages.
Biplanes were popular in WWI not because of any inherent advantage, but simply because technology hadn't advanced far enough to allow any other options. Once the monoplane was properly developed, it allowed much higher speeds (450mph for a late WWII fighter, 300mph for an Early WWII fighter, compared to around 120 mph for a late WWI fighter). High speed fighters beat low speed fighters (at least before stealth technology and guided missiles)
There were a few leftover biplanes in WWII - The Fairey Swordfish torpedo bomber was one of the most widely used, as the low speed and good lift from the biplane design allowed it to carry a heavy torpedo, while the drag issues didn't matter at the slow speeds they needed to fly to be able to release the torpedo properly and on target. Others were the Gloster Gladiator (UK), Fiat CR.42 (Italy) and Polikarpov I-15 (Soviet Union) which achieved varying levels of success in the early stages, but were well and truly outclassed when facing even early-war monoplanes.
This answer on a similar question gives some useful context about the direct advantages and disadvantages. It doesn't directly explain why biplanes fell out of favour, but it may help understand my own answer here.