Suppose F(X,Y) is a homogeneous polynomial with integer coefficients that is irreducible over the rational numbers and has degree at least 3. Fix an integer m. We will discuss integer solutions x and y to the Thue equation F(x,y)=m. We will go over some history of efforts to understand such solutions along with conjectures both old and new(er). We'll demonstrate a geometric approach which gives some insight into a deep conjecture of Stewart regarding the number of possible solutions. No previous knowledge of Diophantine equations/inequalities will be assumed; the talk should be accessible to anyone with an understanding of undergraduate mathematics.