Plasma: Free charged particles, usually in equal numbers, such as ions and electrons. Unlike gases, plasma may self-generate magnetic fields and electric currents and respond strongly and collectively to electromagnetic forces. Plasma is very uncommon on Earth (except for the ionosphere), although it is the most common state of matter in the universe.
Supercritical fluid: At sufficiently high temperatures and pressures, the distinction between liquid and gas disappears.
Degenerate matter: matter under very high pressure, supported by the Pauli exclusion principle.
Electron-degenerate matter: found inside white dwarf stars. Electrons remain bound to atoms but can transfer to adjacent atoms.
Neutron-degenerate matter: found in neutron stars. Vast gravitational pressure compresses atoms so strongly that the electrons are forced to combine with protons via Inverse beta decay, resulting in a super dense conglomeration of neutrons. (Normally free neutrons outside an atomic nucleus will decay with a half-life of just under fifteen minutes, but in a neutron star, as in the nucleus of an atom, other effects stabilize the neutrons.)
Strange matter: A type of quark matter that may exist inside some neutron stars close to the Tolman–Oppenheimer–Volkoff limit (approximately 2–3 solar masses). May be stable at lower energy states once formed.
Quantum spin Hall state: a theoretical phase that may pave the way for developing electronic devices that dissipate less energy and generate less heat. This is a derivation of the quantum Hall state of matter.
Bose–Einstein condensate: a phase in which a large number of bosons all inhabit the same quantum state, in effect becoming one single wave/particle. This is a low-energy phase that can only be formed in laboratory conditions and at very low temperatures. It must be close to zero kelvin, or absolute zero. Satyendra Nath Bose and Albert Einstein predicted the existence of such a state in the 1920s, but it was not observed until 1995 by Eric Cornell and Carl Wieman.
Fermionic condensate: Similar to the Bose-Einstein condensate but composed of fermions, also known as Fermi-Dirac condensate. The Pauli exclusion principle prevents fermions from entering the same quantum state, but a pair of fermions can behave like a boson, and multiple such pairs can then enter the same quantum state without restriction.
Superconductivity: is a phenomenon of exactly zero electrical resistance and expulsion of magnetic fields occurring in certain materials when cooled below a characteristic critical temperature. Superconductivity is the ground state of many elemental metals.
Superfluid: A phase achieved by a few cryogenic liquids at extreme temperature at which they become able to flow without friction. A superfluid can flow up the side of an open container and down the outside. Placing a superfluid in a spinning container will result in quantized vortices.
Supersolid: similar to a superfluid, a supersolid can move without friction but retains a rigid shape.
Quantum spin liquid: A disordered state in a system of interacting quantum spins which preserves its disorder to shallow temperatures, unlike other disordered states.
String-net liquid: Atoms in this state have unstable arrangements, like a liquid, but are still consistent in the overall pattern, like a solid.
Time crystals: A state of matter where an object can have movement even at its lowest energy state.
Rydberg polaron: A state of matter that can only exist at ultra-low temperatures and consists of atoms inside of atoms.
Black superionic ice: A state of matter that can exist under very high pressure while excited by super lasers.
Beyond the classical states of matter, what are the modern states of matter?
1. Plasma: Plasma is the most common state of matter in the universe, but it is uncommonly found on Earth. This state of matter is comprised of free charged particles, such as ions and electrons. Similar to a gaseous state, however, it may generate a magnetic field. 
2. Supercritical fluid: Matter that lacks a clear distinction between being a gas and a liquid. This state is found at high temperatures and pressures.
3. Degenerative Matter: A highly dense state of matter due to very high pressures. This state of matter is naturally found in stars in their final evolutionary states. 
4. Bose-Einstein condensate: A phase of matter in which a large number of bosons all inhabit the same quantum state. This phase requires temperatures near absolute zero of very low density gas.
5. Fermionic condensate: Similar to Bose-Einstein condensate, except with fermions instead of bosons. 
6. Superconductivity: Certain elemental materials may achieve a ground state of superconductivity where there is no electrical resistance and magnetic flux fields are expelled. While most conductors will have reduced resistance as the temperature is lowered, superconductors will abruptly achieve zero resistance at a certain critical temperature. 
7. Superfluid: Some fluids may enter this state of matter where, at extremely low temperatures, they become able to flow without friction.
8. Supersolid: Similar to a superfluid, except it can move without friction in a rigid, solid shape.
9. Quantum spin liquid: A state of matter found in certain materials related to the quantum spin. In most magnetic materials, spins tend to form patterns based on how they interact with other neighboring spins. In a quantum spin liquid, however, they remain in a disordered state, never forming a pattern that prevents it from reaching a stable, ordered state.
10. String-net liquid: Atoms in this state have a consistent overall pattern like a solid, but their arrangement is unstable.
11. Time crystals: A quantum system of particles that exhibits non-kinetic motion of particles at their lowest-energy state. 
12. Rydberg polaron: An exotic state of matter created at low temperatures in which a very large atom contains other ordinary atoms between the nucleus and electrons. 
13. Black superionic ice: This matter is a phase of water that exists at extremely high pressures and temperatures.