  We give a simplified proof of the Berger-Coburn theorem on the boundedness of
Toeplitz operators and extend this theorem to the setting of $p$-Fock spaces
$(1\leq p \leq \infty)$. We present an overview of recent results by various
authors on the compactness characterization via the Berezin transform for
certain operators acting on the Fock space. Based on these results we present
three new characterizations of the Toeplitz $C^*$ algebra generated by Toeplitz
operators with bounded symbols.
