This is a review of the authors' recent results on an integrable structure of the melting crystal model with external potentials. The partition function of this model is a sum over all plane partitions (3D Young diagrams). By the method of transfer matrices, this sum turns into a sum over ordinary partitions (Young diagrams), which may be thought of as a model of $q$-deformed random partitions. This model can be further translated to the language of a complex fermion system. A fermionic realization of the quantum torus Lie algebra is shown to underlie therein. With the aid of hidden symmetry of this Lie algebra, the partition function of the melting crystal model turns out to coincide, up to a simple factor, with a tau function of the 1D Toda hierarchy. Some related issues on 4D and 5D supersymmetric Yang–Mills theories, topological strings and the 2D Toda hierarchy are briefly discussed.