It seems that Lars Brünjes and Christian Serpé have a whole program for introducing non standard mathematics in algebraic geometry; they wish to play with non standard contructions, seen internally and externally (the interest of this game consists precisely to look at the non-classical logic (or internal) point of view and at the classical (or external) one at the same time). For instance, for an infinite prime number $P$, the ring $\mathbf{Z}/P\mathbf{Z}$ behaves internally like a finite field, while externally, it is a field of characteristic zero which contains an algebraic closure of $\mathbf{Q}$. Non-standard constructions can often be interpreted in a precise way as standard ones using ultraproducts and ultrafilters. Their purpose is to develop all the tools of classical algebraic geometry (homotopical and homological algebra, stacks, étale cohomology, algebraic K-theory, higher Chow groups...) in a non-standard way, in order to prove facts in the classical setting. Most of their papers can be found here (their papers contains more precise ideas on the possible interpretations and explanations). Brünjes and Serpé see non-standard mathematics as an enlargement of standard mathematics, and their work deals a lot in making this precise. However, they seem to have quite few concrete problems in mind. For instance, they have found sufficient conditions on cohomology classes to be algebraic in a very classical sense (see arXiv:0901.4853).
(C)
Recently applied model theorists have touched many areas of algebra, algebraic geometry, number theory and even analysis structures.
(1) Exponential fields:
Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s:
Given any $n$ complex numbers $z_1,\dots,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$, the extension field $\mathbb{Q}(z_1,\dots,z_n, \exp(z_1),\dots,\exp(z_n))$ has transcendence degree of at least $n$ over $\mathbb{Q}$.
In 2004, Boris Zilber systematically constructs exponential fields $K_{\exp}$ that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinal. Zilber axiomatises these fields and by using the Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proves that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. See here and here for more.
(2) Polynomial dynamics:
The connection between algebraic dynamics and the model theory of difference fields was first noticed by Chatzidakis and Hrushovski. A series of three papers entitled "Difference fields and descent in algebraic dynamics".
It seems that the first-order theories of algebraically closed difference fields where the automorphism is "generic" are quite nice. See here for more result by Scanlon and Alice Medvedev.
(3) Diophantine geometry:
Hrushovski, Scanlon and their students have worked on model theory and its application in Diophantine geometry.
See here for information about applications of model theory in Diophantine geometry.
(4) Algebraic geometry:
The Mordell-Lang conjecture for function fields: Let $k_0\subset K$ be two distinct algebraically closed fields. Let $A$ be an abelian variety defined over $K$, let $X$ be an infinite subvariety of $A$ defined over $K$ and let $\Gamma$ be a subgroup of "finite rank" of $A(K)$. Suppose that $X\cap \Gamma$ is Zariski dense in $X$ and that the stabilizer of $X$ in $A$ is finite. Then there is a subabelian variety $B$ of $A$ and there are $S$, an abelian variety defined over $k_0$, $X_0$ a subvariety of $S$ defined over $k_0$, and a bijective morphism $h$ from $B$ onto $S$, such that $X=a_0 + h^{-1}(X_0)$ for some $a_0$ in $A$.
This theorem is proved by Hrushovski in 1996, see here. For more see this book.
(5) Number theory:
For example see the recent works of Jonathan Pila.
(6) Analysis:
Traditionally model theory is consistent with algebra. But recently, model theorists have been interested in continuous structures that appears in analysis, for example Banach spaces. For more see here.
Model theory has many other application in other fields of mathematics, such as geometric group theory, differential algebra, Berkovich spaces (see recent works of Hrushovski, Loeser, Poonen here and here), approximate groups, etc. (for more see here, here, here and here )
Note: Model theorists have many important and interesting problems in their fields and I believe that the goal of model theory is not necessary to solve the problems of the other fields!
Best Answer
A Fraïssé construction lies at the heart of the proof of my embedding theorems.
Theorem. (J. D. Hamkins, Every countable model of set theory embeds into its own constructible universe, JML 13(2), 2014).
For any two countable models of set theory $\langle M,\in^M\rangle$ and $\langle N,\in^N\rangle$, one of them embeds into the other.
Indeed, a countable model of set theory $\langle M,{\in^M}\rangle$ embeds into another model of set theory $\langle N,{\in^N}\rangle$ if and only if the ordinals of $M$ order-embed into the ordinals of $N$.
Consequently, every countable model $\langle M,\in^M\rangle$ of set theory embeds into its own constructible universe $\langle L^M,\in^M\rangle$.
Furthermore, every countable model of set theory embeds into the hereditary finite sets $\langle \text{HF},{\in}\rangle^M$ of any nonstandard model of arithmetic $M\models\text{PA}$. Indeed, $\text{HF}^M$ is universal for all countable acyclic binary relations.
The sense of embedding here is $j:M\to N$ for which $x\in y\iff j(x)\in j(y)$, which is the model-theoretic sense of embedding, and this is weaker than those usually considered in set theory because they need not be elementary nor even $\Delta_0$-elementary. An embedding $j:M\to N$ is simply an isomorphism of $M$ with a substructure of $N$, not necessarily transitive.
The proof proceeds by finding sufficiently universal substructures of any model of set theory, using essentially a Fraïssé limit construction, as you can see in the paper. Basically, this construction shows that every countable model of set theory $\langle M,\in^M\rangle$ has a submodel that is universal for all countable $\text{Ord}^M$-graded digraphs. A grading of a digraph is a linear pre-order on that digraph, such that the linear order strictly increases when following any edge of the digraph. Any model of set theory $\langle M,\in^M\rangle$ admits an $\text{Ord}^M$ grading by means of the von Neumann rank, since whenever $x\in y$, then the rank of $y$ strictly exceeds that of $x$.
It happens that I am speaking today on this topic at the CUNY set theory seminar, in about an hour.