  Let $\mathcal{L}\subseteq \mathcal{L}^{\prime }$ be first order languages,
let $R\in \mathcal{L}^{\prime }-\mathcal{L}$ be a relation symbol, and let $%
\mathcal{K}$ be a class of $\mathcal{L}^{\prime }$-structures. In this paper we
present semantical conditions equivalent to the existence of an
$\mathcal{L}$-formula $\varphi \left( \vec{x}\right) $ such that
$\mathcal{K}\vDash \varphi (\vec{x})\leftrightarrow R(\vec{x})$, and $\varphi $
has a specific syntactical form (e.g., quantifier free, positive and quantifier
free, existential horn, etc.). For each of these definability results for
relations we also present an analogous version for the definability of
functions. Several applications to natural definability questions in universal
algebra have been included; most notably definability of principal congruences.
The paper concludes with a look at term-interpolation in classes of structures
with the same techniques used for definability. Here we obtain generalizations
of two classical term-interpolation results: Pixley's theorem for quasiprimal
algebras, and the Baker-Pixley Theorem for finite algebras with a majority
term.
