The classification of countable models up to isomorphism plays a critical role in many open problems in logic. However, just using first order logic, there are models that agree on all formulas, but are not isomorphic. Using infinitary logic, we can circumvent this problem. In this talk, we show Scott's Isomorphism Theorem, which states that two countable models are isomorphic if and only if they satisfy the same $L_{\omega_1, \omega}$ sentence.