Weighted Bergman kernels and mathematical physics

We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Berman kernels of domains $Omega subset {mathbb C}^n$ but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold and turn out to be an efficient computational tool, useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version [for weights of the form $gamma = | varphi |^m$ on strictly pseudoconvex domains $Omega = { varphi < 0 } subset {mathbb C}^n$] of Fefferman's asymptotic expansion of the Bergman kernel and discuss its possible extensions [to more general classes of weights] and implications, e.g. such as related to the construction and use of Fefferman's metric (a Lorentzian metric on $partial Omega X S^1$). Several open problems are indicated throughout the survey.