We develop a statistical mechanical framework for the folding thermodynamics of pseudoknotted structures. As applications of the theory, we investigate the folding stability and the free energy landscapes for both the thermal and the mechanical unfolding of pseudoknotted chains. For the mechanical unfolding process, we predict the force-extension curves, from which we can obtain the information about structural transitions in the unfolding process. In general, a pseudoknotted structure unfolds through multiple structural transitions. The interplay between the helix stems and the loops plays an important role in the folding stability of pseudoknots. For instance, variations in loop sizes can lead to the destabilization of some intermediate states and change the (equilibrium) folding pathways (e.g., two helix stems unfold either cooperatively or sequentially). In both thermal and mechanical unfolding, depending on the nucleotide sequence, misfolded intermediate states can emerge in the folding process. In addition, thermal and mechanical unfoldings often have different (equilibrium) pathways. For example, for certain sequences, the misfolded intermediates, which generally have longer tails, can fold, unfold, and refold again in the pulling process, which means that these intermediates can switch between two different average end-end extensions.