Just to start off the discussion, did you know that time does not flow at a constant rate everywhere in the universe?  It's true. Einstein first theorized this in his Theory of Special and General Relativity, and it's been proven multiple times by various experiements.  GPS satellites, in fact, have to compensate because time passes more slowly on Earth's surface than it does in high orbit due to the influence of Earth's gravity.  Without such clock adjustments, the GPS system would begin to drift further and further out of alignment.

 

The effect is very small around Earth.  It takes a huge gravity field produced by a super-massive object such as a black hole (or extremely high velocities near the speed of light) to cause enough time slow-down (what is more precisely known as time dilation) to be noticable by human beings.  And you probably still wouldn't notice it because everything around you would be slowed down the same way and things would still seem normal.

 

Einstein's formulas tell us that time continues to slow down as long as the necessary conditions become more and more acute.  A black hole's event horizon is that place where condtions are so extreme that time actually stops, according to the equations.  The problem is, just at that point, the equations break down and we can not be sure what is actually happening right at the critical point.  It appears a new kind of science may be needed to explain the universe under these very extreme and strange conditions.  Quantum Gravity is the moniker for the scientific theory with the most promise for explaining these mysterious occurances in our universe.