The dimer model is the study of random perfect matchings on graphs, and has a long history in statistical mechanics. On the hexagonal lattice it is equivalent to tilings of the plane by lozenges and to 3D stepped surfaces called skew plane partitions - 3 dimensional analogues of Young diagrams with a partition removed from the corner. We will discuss the scaling limit of the model under a certain family of measures called "volume"-measures, the limit-shape phenomenon in this model (a form of the law of large numbers), the effects of varying the boundary conditions on the limit shape, and the nature of local fluctuations in various regions of the limit shape. We will also discuss the behaviour of the system when the measure is modified in certain ways.