Consider Young's slits experiment. It consists of a card with two vertical strips in it and a light is projected through it onto black card behind it. Most people expect to see two slits projected onto the blank card behind it, but instead zebra stripes appear, because the light is a wave and it interferes with other light sources, like ripples on a pond. This happens because when the two light waves meet, some are opposite (ie a peak and a trough) and cancel each other out and some are the same and reinforce each other.
Now imagine performing this test with a stream of neutrons. There are only two choices and the neutrons pass through one or other of the slits. As a result there are only two "flashes" on the card. Neutrons do now act like waves, and so the interference pattern is not observed. However, if you fire one neutron at a time, there are still two flashes. The conclusion; the neutron passes through both slits at the same time. Read that bit again.
Why this happens was a mystery for a long time, but there is a solution. The Heisenburg Uncertainty Principle states that it is impossible to know a sub-atomic particles position and vector at the same time. So the neutron's position cannot be determined with any accuracy, because this destroys the data, ie the position is changed by measuring it. So the neutron's position is described by a probability wave function, and this allows it to pass through both slits, as there is a probability that it could pass through either. The neutron exists in a superposition state, where it can occupy two locations at the same time.
To think of it another way, the neutron is too small to be "seen" because it is smaller that the wavelength of light and light simply passes it without reflecting off it. You can determine it's position by using energy with shorter and shorter wavelengths, like Gamma rays, but the shorter wave length means the wave has more energy. When the wave hits the neutron, it transfers part of that energy to the neutron, which makes it change position.* You can use longer wave lengths to measure it, which will transfer less energy to it, but they won't give you as accurate a position. This is where the probability wave function comes in. It provides a range of positions that the neutron can be in. An explanation is available at the Wave-Particle Duality entry.
In Young's experiment, the neutron passes through both slits because there is no wave trying to determine it's position, so it can remain in it's superposition state and create two flashes.
Recently quantum physists have been having some sucess with looking at Hardy's paradox, using a technique called "weak measurement", which lets them look at quantum states without disrupting them. While not being Young's slit experiment, it is peripherally related, as is Shroedinger's cat.
This "weak measurement" using some very strange effects on the quantum level, and while you would think it would collapse the superposition, it doesn't seem to do that. It works by using a beam splitter and making it follow two paths, then re-combining it at a dectector.
Another problem is that entanglement has been observed to act across time as well as space. In particular, when the same particle stream is measured, sometimes once, sometimes twice, in a time frame such that the second measurement affects particles which have already been measured.
Then when the first result for each particle is analysed, you can spot those that were measued twice from the data.
This no more fits the theories than the instant action at a distance that you get with ordinary entanglement, but has the extra problem that it seems to defy causality.
For more inforamtion, look at University of Missouri - St.Louis.