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We initiate a study of asymptotic Toeplitz operators on the Hardy space$H^2(\mathbb{D}^n)$ (over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$).We also study the Toeplitz operators in the polydisc setting. Our main resultson Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let$T_{z_i}$ denote the multiplication operator on $H^2(\mathbb{D}^n)$ by the$i^{th}$ coordinate function $z_i$, $i =1, \ldots, n$, and let $T$ be a boundedlinear operator on $H^2(\mathbb{D}^n)$. Then the following hold:  (i) $T$ is a Toeplitz operator (that is, $T = P_{H^2(\mathbb{D}^n)}M_{\varphi}|_{H^2(\mathbb{D}^n)}$, where $M_{\varphi}$ is the Laurent operatoron $L^{2}(\mathbb{T}^n)$ for some $\varphi \in L^\infty(\mathbb{T}^n)$) if andonly if $T_{z_i}^* T T_{z_i} = T$ for all $i = 1, \ldots, n$.  (ii) $T$ is an asymptotic Toeplitz operator if and only if $T =\mbox{~Toeplitz} + \mbox{~compact}$.  The case $n = 1$ is the well known results of Brown and Halmos, and Feintuch,respectively. We also present related results in the setting of vector-valuedHardy spaces over the unit disc.

Toeplitz and Asymptotic Toeplitz operators on $H^2(\mathbb{D}^n)$