In 1985, Gromov discovered a rigidity phenonmenon for symplectic embeddings which led to the concept of Gromov width: a measure of the largest ball that can be symplectically embedded inside a symplectic manifold. This is an invariant of the symplectic form. The central theme of this thesis is computing the Gromov width of symplectic cuts. We do this for two classes of symplectic manifolds: four-dimensional toric manifolds and complex Grassmannian manifolds. Symplectic cutting results in symplectic manifolds of smaller volume. A natural question is whether the Gromov width decreases (or at least does not increase) under this operation. In this thesis we use symplectic embedding techniques and theory of J-holomorphic curves to establish lower and upper bounds on Gromov width. In the case of 4-dimensional toric symplectic manifolds, we answer the question positively for any symplectic cuts that result in smooth manifolds. We then compute the exact Gromov width of certain cuts of complex Grassmannians, again establishing the desired monotonicity.