(What follows has only to do with line of sight. Cover works differently. Line effects [such as lightning bolt] also work differently. See the rulebook for details.)
For players of both D&D Minis and Star Wars Minis, there is a common misconception that line of sight is determined by tracing an imaginary line from any corner of one creature's space to any corner of another creature's space. In fact, as of this writing, the official FAQs for those games incorrectly reinforce the misconception.
The actual rules for both games tell the truth: Line of sight is determined by tracing an imaginary line from any part of one creature's space to any part of another creature's space; you don't have to use corners. In fact, you can't just use corners -- if you do, you may miss an existing line of sight. To ensure you don't miss one, you must sometimes go from a non-corner part of one creature's space to a non-corner part of another creature's space. Here's why:
In the above situation, two creatures -- one yellow and one greenish -- are near walls. The battle grid is represented by the gray lines, and walls are represented by black lines. This particular situation is most easily reproduced by the Treasure Room feature tile for D&D Minis, but many other similar situations exist in both games.
Before continuing, it is important to review the rule for determining whether an imaginary line provides line of sight: In order to provide line of sight, an imaginary line must neither cross nor touch a wall. The last part is most important; if the imaginary line even just touches a wall at a single point, it doesn't provide line of sight.
If you try to establish line of sight by tracing an imaginary line from the green creature's bottom left corner to any of yellow's corners, you get this:
Clearly, all four of those lines touch walls at the starting point and/or at other places, so none of them provide line of sight. Obviously, the same thing happens if you were to try to trace line of sight from green's bottom right corner.
If you try to establish line of sight from green's top left or top right corner to any of yellow's corners, you get this:
All of those imaginary lines touch and/or cross walls, so none of them provide line of sight.
Trying to establish line of sight from yellow's top right corner looks like this:
Again, no success, since all of those lines cross or touch walls. You get the same results if you try to trace from yellow's top left corner.
If you try to establish line of sight from yellow's bottom left or bottom right corner to any of green's corners, you get this:
Once again, no success. All of those lines cross or touch walls.
However, yellow and green do have line of sight to one another. The only way to see that is by tracing a line from a non-corner part of yellow's space to a non-corner part of green's space:
That particular imaginary line starts within one creature's space and ends in the other creature's space, and doesn't cross or touch a wall. Therefore, the two creatures have line of sight to one another despite the fact that you cannot trace a line from one creature's corner to the other creature's corner.
Now I'm off to see whether I can get the FAQs for both games fixed... :)
Also: Thanks to dagni and Tried for showing me this easy-to-reproduce situation at Gen Con Indy, 2004!