This paper concerns algebraic and spectral properties of Toeplitz operators
T
φ
T_{\varphi }
, on the Hardy space
H
2
(
T
)
H^{2}({\mathbb {T}})
, under certain assumptions concerning the symbols
φ
∈
L
∞
(
T
)
\varphi \in L^{\infty }({\mathbb {T}})
. Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at
T
φ
T_{\varphi }
, for each quasicontinuous
φ
\varphi
. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.