This Q&A is intended as a canonical(-ish) concerning two-dimensional (and multi-dimensional) interpolation using scipy. There are often questions concerning the basic syntax of various multidimensional interpolation methods, I hope to set these straight too.

I have a set of scattered two-dimensional data points, and I would like to plot them as a nice surface, preferably using something like contourf or plot_surface in matplotlib.pyplot. How can I interpolate my two-dimensional or multidimensional data to a mesh using scipy?

I've found the scipy.interpolate sub-package, but I keep getting errors when using interp2d or bisplrep or griddata or RBFInterpolator (or the older Rbf). What is the proper syntax of these methods?