Harmonic grid theory
You can use a Tonnetz to quickly look up what notes are in a specific key/scale, look up what notes are in a specific chord, analyze and make sense of chord progressions, compose interesting chord progressions in an intuitive way, find the relative minor or major to a key, easily transpose melodies or chord progressions and many more.
Really, the only limit here is one's own imagination.
So how does all of this work?
This video describes the Tonnetz and voice leading. Harmonic motion between chords can easily be accomplished by moving around the Tonnetz in three different ways .
Harmonic analysis of the song Take A Bow composed by Matthew Bellamy performed by Muse. Chords are represented within the pitch space named as Tonnetz, which displays musical pitches along axis associated with the intervals of fifth (horizontal), minor and major thirds (diagonals). Among other notable properties, the Tonnetz makes appear minor and major triads as triangles which facilitates the visualization of tonal chord progressions.
A visualization of Gymnopédie No. 1 by Erik Satie