A symplectic Lie groupoid is a Lie groupoid with a multiplicative symplectic form. Taking the perspective that such an object is symplectic manifold with an extra categorical structure, it is natural to apply the machinery of Floer theory; the extra structure is then expected to yield a monoidal structure on the Fukaya category, and new operations on the closed string invariants. I will take an examples-based approach to working out what these structures are, focusing on cases where the Floer theory is tractable, such as the cotangent bundle of a compact manifold. I will begin with a review of the Floer theoretic constructions I will use, so knowledge of Floer Theory and Fukaya categories will not be assumed.