Compositional Quantum Field Theory (CQFT) is an axiomatic framework for reasoning about Quantum Field Theory, based on the principles of locality and compositionality. The first ingredient of CQFT is a spacetime gluing syntax based on gluing maps and gluing diagrams. I will present this in detail. Once this is set up, I will introduce CQFT axiomatically, and I will show how to recast these axioms as a type of functorial semantics translating our spacetime syntax into Hilbert spaces, using algebras over *-operads and constrained maps between them. Finally, we will investigate how in CQFT in dimension 2, e.g. in 2 dimensional Quantum Yang-Mills theory, our functorial semantics viewpoint allows for certain structures to emerge from first principles. An example of this is the structure of cyclic, involutive A_oo-algebra on the state space of the interval.