Using a new theoretical model, we explore the conformational energy landscapes and the statistical thermodynamics of RNA secondary structure folding. The model is physical, treats the conformational entropies and excluded volume explicitly, and has previously been shown to give reasonable agreement with experimental melting curves. Here we use it to design energy landscapes that can be tested by experiments. The model predicts that even simple RNA's, less than 60 nucleotides long and having only secondary structures, can have remarkably complex and bumpy landscapes that can be altered substantially by small mutations. Moving a GC block around in a sequence of fixed composition can lead from one melting peak to two. A mutation can switch it back to one. Two melting peaks do not imply two simple stem melting events. In one system, we find 5 transitions, each of which can be either 2-state or 1-state. In some sequences we find triple points, where 3 macroscopically identifiable species have equal populations at the same temperature. We show designs of RNA's with multiple native states, and conformational switching between them, as functions of mutations or temperature.