Since the discovery of the first L dwarf 19 years ago and the discovery of the first T dwarf 7 years after that, we have amassed a large list of these objects, now numbering almost six hundred. Despite making headway in understanding the physical chemistry of their atmospheres, some important issues remain unexplained. Three of these are the subject of this paper:(1) What is the role of ``second parameters'' such as gravity and metallicity in shaping the emergent spectra of L and T dwarfs? Can we establish a robust classification scheme so that objects with unusual values of log(g) or [M/H], unusual dust content, or unresolved binarity are easily recognized? (2) Which physical processes drive the unusual behavior at the L/T transition? Which observations can be obtained to better constrain the problem? (3) What will objects cooler than T8 look like? How will we know a Y dwarf when we first observe one?