What are some interesting uses for/motivations of projective spaces? I have trouble motivating myself to think about real projective spaces, for instance. Are there any cool results about them? Are there any motivating examples?
 A: An excellent motivation is artistic/biologic. When an artist try to represent the reality on a painting, he must project it on a plane. The geometry needed to make the painting as close as possible to the reality is  the projective geometry. To represent tiles keeping the proportions, one need to use the cross-ratio. Apparently this is a mathematician (Luca Pacioli) who  learned this to A Durer.
More recently this geometry is now used in the shape recognition problem. From 2 or 3 picture, how can a computer re-build the three dimensional object.(see here for a nice introduction http://robotics.stanford.edu/~birch/projective/projective.pdf)
A: Since you included differential-geometry as one of the tags, one could mention a pretty result called Pu's theorem about the real projective plane $RP^2$ with an arbitrary Riemannian metric.  Consider the least length $L$ of a noncontractible loop in $RP^2$.  Let $A$ be the total area of the metric on $RP^2$.  Then these two invariants are related by a kind of "reverse isoperimetric inequality", namely $L^2 \leq \frac{\pi}{2}A$. This inequality is sharp, in the sense that the boundary case of equality is attained by a "round" metric, i.e., a metric of constant Gaussian curvature.
