Not sure of any of the theory behind it but Melodyne's is probably the best I've ever worked with.

I second this, though I haven't tried the alternatives which I hear are also good. There is some value to specific kinds of pitch shifters, like the Boss Octave pedal, the Digitech Whammy pedal, and Eventide's harmonizers. Depends on what you are trying to do and what kind of control you want.

“Elastique” time stretch algorithm by zplane (0) seems to be a common choice by plug-in companies. Their C++ library is pretty straightforward to use, and most importantly it sounds good.

Generally speaking, phase vocoder (1) algorithms are what you’d want for developing your own pitch shifting algorithm.

[0]https://licensing.zplane.de/

[1]https://cmtext.indiana.edu/synthesis/chapter4_pv.php

Serato Pitch n Time is used on master mixes to correct wholesale tuning issues.

but really depends on what you're after (creative or precision)