A general formulation is given for the concepts of quasi-bounded and singular functions, thereby extending to a much broader class of functions the concepts initially formulated by Parreau in the harmonic case. Let
Ω
\Omega
be a bounded Euclidean region. With the underlying space taken as the class
M
\mathcal {M}
of all nonnegative functions u on
Ω
\Omega
admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over
λ
≥
0
\lambda \geq 0
of the regularized reduced functions for
(
u
−
λ
)
+
{(u - \lambda )^ + }
. Quasi-bounded and singular functions are then defined as those u for which
S
u
=
0
Su = 0
and
S
u
=
u
Su = u
, respectively. A development based on properties of the operator S leads to a unified theory of quasi-bounded and singular functions, correlating earlier work of Parreau (1951), Brelot (1967), Yamashita(1968), Heins (1969), and others. It is shown, for example, that a nonnegative function u on
Ω
\Omega
is quasi-bounded if and only if there exists a nonnegative, increasing, convex function
φ
\varphi
on
[
0
,
∞
]
[0,\infty ]
such that
φ
(
x
)
/
x
→
+
∞
\varphi (x)/x \to + \infty
as
x
→
∞
x \to \infty
and
φ
∘
u
\varphi \circ u
admits a superharmonic majorant. Extensions of the theory are made to the vector lattice generated by the positive cone of functions u in
M
\mathcal {M}
satisfying
S
u
≤
u
Su \leq u
.