Model Theory, as a subject, has grown tremendously over the last few decades. Starting out as a branch of mathematical logic, 
 it has now wide applications in most branches of mathematics, with algebra, algebraic geometry, number theory,  combinatorics and 
 even analysis to name a few. In this talk, I will give a brief introduction to model theory, talk about the compactness theorem 
 (one of the main tools in model theory) and how it is used, and give one famous  application of model theory to algebra, namely, 
 the Ax-Kochen Theorem, the answer to Artin's Conjecture. Time permitting, I will talk about a few more recent results in this 
 direction.