  Majorana fermions are the real (in a mathematical sense) counterparts of
complex fermions like ordinary electrons. The promise of topological quantum
computing has lead to substantial experimental progress in realizing these
particles in various synthetic platforms. The realization of Majorana fermions
motivates a fundamental question: What phases of matter can emerge if many
Majorana fermions are allowed to interact? Here we review recent progress in
this direction on the proposed experimental setups, analytical and numerical
results on low-dimensional lattice models, and the exactly solvable
Sachdev-Ye-Kitaev model. The early progress thus far suggests that strongly
correlated phases of matter with Majorana building blocks can exhibit many
novel phenomena, such as emergent spacetime supersymmetry, topological order
and the physics of black holes, in condensed matter systems. They may also
provide alternative avenues for universal topological quantum computing through
the realization of Fibonacci phase and measurement-based only surface codes.
