natural gradient descent is big in reinforcement learning -- proximal policy optimization (popular RL algorithm, most famously employed for OpenAI DotA bot) can be thought of as a cheap approximation to it. people also use something called KFAC, another approximation of natural gradient.

you can understand everything in those papers without understanding differential geometry, just basic linear algebra -- the core idea is just preconditioning gradients by the inverse of the Fisher information matrix.

the deeper theoretical stuff, i have no idea about, it seems like it must be fascinating though!