When creating things like physics in games you need an integrator. I've seen Verlet integration mentioned several places as a great alternative to Euler integration. For instance in the [famous document by Thomas Jakobsen](http://web.archive.org/web/20100111035201/http://www.teknikus.dk/tj/gdc2001.htm). However in [this article](http://gafferongames.com/game-physics/integration-basics/) Glenn Fiedler writes: 

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> Rather than introduce you to the vast array of different integrators that exist, I’m going to cut to the chase and go straight to the best. This integrator is called the Runge Kutta order 4 integrator aka RK4.
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So apparently there's no silver bullet. What is the pros and cons of different integrators? Regarding simplicity, speed, accuracy, stability, etc. Which integrators is most suitable for which types of games? When would you use Verlet, RK4 or others? Should you ever use Euler?