  We present a general algorithm for solving all two-variable polynomial
Diophantine equations consisting of three monomials. Before this work, even the
existence of an algorithm for solving the one-parameter family of equations
$x^4+axy+y^3=0$ has been an open question. We also present an elementary method
that reduces the task of finding all integer solutions to a general
three-monomial equation to the task of finding primitive solutions to equations
with three monomials in disjoint variables. We identify a large class of
three-monomial equations for which this method leads to a complete solution.
Empirical data suggests that this class contains $100\%$ of three-monomial
equations as the number of variables goes to infinity.
