  We study the restriction operator from the Bergman space of a domain in
$\mathbb{C}^n$ to the Bergman space of a non-empty open subset of the domain.
We relate the restriction operator to the Toeplitz operator on the Bergman
space of the domain whose symbol is the characteristic function of the subset.
Using the biholomorphic invariance of the spectrum of the associated Toeplitz
operator, we study the restriction operator from the Bergman space of the unit
disc to the Bergman space of subdomains with large symmetry groups, such as
horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp
estimate of the norm of the restriction operator in case the domain and the
subdomain are balls. We also study various operator theoretic properties of the
restriction operator such as compactness and essential norm estimates.
