We introduce the notion of a connected sum of graded Artinian Gorenstein algebras following the approach of Ananthnarayan, Avramov, and Moore, who studied this construction for local rings. Our connected sums can be viewed as an algebraic analogue for the topological construction of the same name. We give an alternate description of this construction in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property, thus providing new classes of rings which satisfy these properties. This is based on joint work with Tony Iarrobino and Chris McDaniel.