A Fukaya category of a symplectic manifold has objects as certain Lagrangian submanifolds with a certain A-infinity category structure. In this talk, we take on the arduous task of attempting to define a Fukaya category. We will begin with some basic symplectic geometry leading to motivation coming from Lagrangian Floer (co)homology. We then work to understand A-infinity categories before finally tackling Fukaya categories.