  Let $T(X)$ be the full transformation monoid on a nonempty set $X$. In this
paper, we prove that an element of $T(X)$ is unit-regular if and only if it is
semi-balenced. For a partition $\mathcal{P}$ of $X$, we characterize the
unit-regular elements in monoid $T(X, \mathcal{P}) = \{f\in T(X)|(\forall X_i
\in \mathcal{P}) (\exists X_j \in \mathcal{P})\; X_i f \subseteq X_j\}$ under
composition and its submonoids $\Sigma(X, \mathcal{P}) = \{f\in T(X,
\mathcal{P})|(\forall X_i \in \mathcal{P})\; Xf \cap X_i \neq \emptyset\}$,
$\Gamma(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})|(\forall X_i\in
\mathcal{P})(\exists X_j\in \mathcal{P})\; X_i f = X_j\}$,
$\Omega(X,\mathcal{P}) = \{f\in T(X, \mathcal{P})|(\forall x, y\in X, x\neq
y)(x, y)\in E \Longrightarrow xf \neq yf\}$, and $T_{E^*}(X) = \{f\in T(X)| (x,
y)\in E \Longleftrightarrow (xf, yf)\in E\}$, where $E$ is the equivalence
relation induced by $\mathcal{P}$. We also characterize the regular elements in
$\Sigma(X,\mathcal{P})$ and $\Omega(X,\mathcal{P})$.
