Math rock is a relatively new sub-genre of rock music, characterized by complex, atypical rhythms (often including irregular stopping and starting), frequent use of irregular time-signatures, and dissonant, angular guitars. Math rock gained popularity in the mid to late-'90s, when groups like Polvo and Chavez had small but dedicated followings among indie and noise rockers on college campuses. The peak of this genre came in 2 waves, one in the mid-'90s, and one in the dawn of the modern internet, in the mid to late-'00s.
It is related to post-rock, a result of a shared scene of origin. There are differences to be noted between early post-rock and math-rock as well, though: whereas post-rock has distinct jazz influences, math rock is more dense and complex, filled with difficult time signatures and intertwining phrases. The style is a little more 'rockist' than post-rock, since it's usually played by small, guitar-led bands.
While post-rock can be described as pushing the boundaries of rock by using rock instrumentation for non-rock purposes, math rock does not necessarily do this; instead, it tries to push the boundaries of rock by raising the complexity of rock music itself. It generally makes use of complex rythms, dissonant guitar chords, and unconventional playing techniques.
Math Rock can be divided into two periods: a first wave in the early 90's, after genre progenitors Slint, where Polvo, Chavez, and Blume Smells were the most iconic examples. This period had a more noise-rock-influenced, heavy sound. Emerging from the same music scenes (Chicago for example) that spawned early post-rock, there is quite a bit of overlap with that genre. Math rock then faded away, the tag disavowed by bands characterized as such.
The second wave came in the late 00's, when new bands arose with a cleaner, less noisy sound, though with the same complexity and rhythmic playfulness that makes math rock what it is. Examples of this include Maps & Atlases, Giraffes? Giraffes!, Hella, Don Caballero and Tera Melos. These bands are often more stylistically distanced from post-rock.
Lately a few bands have sprung up that try to make math rock more fun and accessible; the most prominent examples may include Damiera and Battles. .