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We investigate problems related with the existence of square integrableholomorphic functions on (unbounded) balanced domains. In particular, we solvethe problem of Wiegerinck for balanced domains in dimension two. We also give adescription of $L_h^2$-domains of holomorphy in the class of balanced domainsand present a purely algebraic criterion for homogeneous polynomials to besquare integrable in a pseudoconvex balanced domain in $\mathbb C^2$. Thisallows easily to decide which pseudoconvex balanced domain in $\mathbb C^2$ hasa positive Bergman kernel and which admits the Bergman metric.

$L_h^2$-functions in unbounded balanced domains