Something that we didn't really go too deeply into, the Jacobian.
The Jacobian is where the r and the r^2sin(theta) com from in the transformation of coordinates from Cartesian to polar and spherical.
Here is the list of Millennium Problems posed by the Clay Mathematics Institute. The first (P vs. NP) is claimed to have been solved, but the jury is still out. The Poincaré Conjecture has been solved, rather famously, by Grigori Perelman a few years ago. The Riemann Hypothesis is probably the most elusive problem in math now that Fermat's Last Theorem and Poincaré have been knocked down.
This helped me when I had trouble visualizing what multi-variable function graphs would look like, without giving away too much like wolfram alpha sometimes does.