This page collects interesting Brainfuck derivatives which take the simple brainfuck language and change something to explore the space of languages near to brainfuck itself. It's aim is to let people flick over a collection of interesting brainfuck derivatives rather than having to dig through a very large list of derivatives. You are welcome to add more! There is a nicely tabulated page of Brainfuck_extensions.
Boolfuck is brainfuck with the cells only being booleans, in this language + and - are replaced with @ which flips the current cell. Despite this is it still turing complete (a simple translation from it to brainfuck is possible by using 8 bitfuck cells to simulate one 8 bit brainfuck cell).
Fuck is a 0 dimensional version of brainfuck with only a single integer cell. The commands < and > are gone. Hello world is still possible but the language is not turing complete.
Gridfuck adds up ^ and down v operations to the language and lets you work with a grid of memory cells rather than a linear tape.
Quantum brainfuck is an enhancement of brainfuck to work on qubits rather than classic integers.
Harmful Brainfuck is a brainfuck derivative that rejects the structured  conditional loop operation and instead has a computed jump operation *. It is still under development.
MNNBFSL is a stack based brainfuck derivative whose turing completeness isn't immediately obvious. It is possible to 'assemble' a higher level version of MNNBFSL that includes jumps down to it though, this shows it Turing complete.
Reversible Brainfuck tweaks a small number of the brainfuck operations so that no information is destroyed and the programs can be run forwards and backwards - despite this it is still Turing complete.
/ˈæmbiːɛf/ is a version of brainfuck that is operates so randomly that its computational class or ability to compute things with is not at all immediately obvious.
Brainhype is a super-turing brainfuck that is able to tell if not only Turing machines halt, but even Brainhype programs too! It is believed to be possible to implement it in scheme-omega (although of course this cannot be done in this universe).
MIBBLLII Isn't brainfuck But Looks Like It Is
MIBBLLII is actually a functional language based on combinatory logic that masquerades as brainfuck. It lets you write out combinators and function application in syntax that looks exactly like a brainfuck program.