I am currently studying the countable Omitting Types Theorem (as presented by Hodges in his book about Model theory):
Let $L$ be a countable first-order language, $T$ a theory in $L$ which has a model, and for each $ m < \omega $ let $\Phi_m$ be an unsupported set over $T$ in $L$. Then $T$ has a model which omits all the sets $\Phi_m$.
Hodges states that the Omitting Types Theorem has different applications but I couldn't find many yet. A professor I talked to also told me about its importance but did not mention anything specific. So I wanted to ask about different applications of it in model theory, but also in other branches of mathematics.