# Solving Diophantine Equations? What are the common techniques?

What does it mean to solve a Diophantine Equation? Some questions are like

Solve $x^2+1=2y^4$ over the integers.

What does it actually mean to solve a diophantine equation? Do we have to find a single solution or an infinite number of them? Can solving mean that we have to prove that none of them exist?

Some Diophantine Equations are very complicated.

Also, if we have to find solutions, what are the techniques involved in solving Diophantine Equations? By techniques, I mean some procedures or ideas to solve them. Which are the common techniques employed while solving them? I need a list of techniques that I can try while solving a Diophantine Equation and I believe that I have a very little knowledge about these techniques.