We present the theory for the conformational free energies of RNA molecules and other polymers that have pseudoknots. We derive an expression for the partition function by counting conformational loops that we call pseudoknot core units, using the theory of self-avoiding walks. We show that the thermal transitions of pseudoknot polymers to their denatured states are not two-state; they are continuous. We also find that the entropy of a pseudoknot loop depends not only on its chain length, as is assumed in most current RNA models, but also on the length of the adjacent stem, because stems are stiff and the stiffness determines the starting and ending points of the loop.