In this thesis, we further develop weighted pluripotential theory. We define admissibility conditions for the weights in the local setting. We obtain many results on weighted relative extremal functions and weighted relative capacities. We prove that the weighted relative capacity of a compact set is less than or equal to the total Monge-Ampère mass of the weighted relative extremal function and we show via an example that this inequality may be strict.

We also define weighted Tchebychev constants and weighted Alexander-Siciak capacities. We show that these are equal for compact subsets of the unit ball of [special characters omitted] and that weighted Tchebychev polynomials exist if the weight is admissible. We obtain some results on weighted equilibrium measures such as the weighted equilibrium measure is supported on the boundary if the weight is superharmonic or maximal plurisubharmonic. We give an example of a compact set to show that weighted extremal measure and weighted relative extremal measure need not be mutually absolutely continuous. We also provide fairly explicit classes of examples of compact sets which arise as the support of weighted equilibrium measures.

We define weighted &thetas;-incomplete extremal functions and advance the &thetas;-incomplete pluripotential theory developed by Callaghan. In particular, we prove a Siciak-Zahariuta type relation between the weighted &thetas;-incomplete extremal function and &thetas;-incomplete polynomials. In addition, we prove a result on strong Bergman asymptotics in the spirit of Berman in this setting.

Finally, we relate the weighted equilibrium measures with the notions of pluriharmonic balayage and boundary measures in the sense of Demailly and Cegrell-Kemppe. This leads to a brief discussion of questions including locally regular points, representing measures, Jensen measures and peak points.