$\newcommand{\nat}{\mathbb{N}}$
$\newcommand{\then}{\ \Longrightarrow\ }$
A partial function $f : \mathbb{N} \to \mathbb{N}$ is said to be $\lambda$-*definable* if there is a term $F \in \Lambda$ such that
$$ \forall n \in \nat: \quad f(m)\!\downarrow = n \then
F c_n =_\beta c_m $$
where $=_\beta$ is beta conversion, and $c_n = \lambda f s. f^n(s)$ is Church's encoding of numerals in $\Lambda$.

It is well-known that every partial computable function is $\lambda$-definable. However, whenever a particular recursive definition of the function $f$ makes use of the $\mu$-operator (minimization), then the $\lambda$-term corresponding to that definition will contain a fixed point combinator, hence not be normalizing in general.

On the other hand, large classes of total recursive functions can be $\lambda$-defined by normalizing terms. For example, Girard showed that every function which is provably total in higher-order arithmetic can be defined by a term typable in the system $F\omega$, and every term in this system is normalizing.