  I survey physics theories involving parallel universes, arguing that they
form a natural four-level hierarchy of multiverses allowing progressively
greater diversity. Level I: A generic prediction of inflation is an infinite
ergodic universe, which contains Hubble volumes realizing all initial
conditions -- including an identical copy of you about 10^(10^29)m away. Level
II: In chaotic inflation, other thermalized regions may have different physical
constants, dimensionality and particle content. Level III: In unitary quantum
mechanics, other branches of the wavefunction add nothing qualitatively new,
which is ironic given that this level has historically been the most
controversial. Level IV: Other mathematical structures give different
fundamental equations of physics. The key question is not whether parallel
universes exist (Level I is the uncontroversial cosmological concordance
model), but how many levels there are. I discuss how multiverse models can be
falsified and argue that there is a severe "measure problem" that must be
solved to make testable predictions at levels II-IV.
