Metamodels are the core of the metamodeling approach and widely used in model driven architecture. The high abstraction level provided by metamodels makes the metamodeling approach popular among software modelers and language designers. However, the metamodeling approach has one big drawback: it does not support instance generation. Instances are particularly important for software modelers and language designers to test or verify their metamodels. Unfortunately, automatically generating metamodel instances is a very challenging task. Furthermore, the generated instances should ideally cover a more generic or specific feature of a metamodel as required by modelers for different purposes. This thesis presents a solution that combines both graph representation and Satisfiability Modulo Theories (SMT) to the problem of metamodel instance generation. The solution consists of two approaches, the first approach presents a new foundation for generating metamodel instances by translating a metamodel to an SMT problem via a bounded graph representation. The second approach investigates generating meaningful metamodel instances by using two new techniques. The first technique generates instances that meet partition-based coverage criteria by using criteria formulas to further constrain the entire generation process. The second technique generates instances that satisfy graph-property based criteria by introducing different scenarios. These two approaches have been prototyped into a new tool, named ASMIG, to demonstrate the feasibility of automatic metamodel instance generation.