A symplectic Lie groupoid is a Lie groupoid with a multiplicative symplectic form. We take the perspective that such an object is symplectic manifold with an extra categorical structure. Applying the machinery of Floer theory, the extra structure is 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.