  In this paper, we give a survey of results obtained recently by the present
authors on real-variable characterizations of Bergman spaces, which are closely
related to maximal and area integral functions in terms of the Bergman metric.
In particular, we give a new proof of those results concerning area integral
characterizations through using the method of vector-valued
Calder\'{o}n-Zygmund operators to handle Bergman singular integral operators on
the complex ball. The proofs involve some sharp estimates of the Bergman kernel
function and Bergman metric.
