Can amorphous solids have energy bands? One can understand the formation of energy bands from the Kronig-Penny model which assumes a periodic potential. But I heard that even if the potential is aperiodic, for example in amorphous substances (glass, plastic) there also exist bands. If not periodicity what is the fundamental physics that causes band formation?
 A: In general, one can understand emergence of band structure starting from two extreme scenarios, both not contradicting formation of broadened levels/bands in amorphous solids:


*

*Start with empty space: electronic dispersion is a parabola due to the kinetic energy of a free electron (or really quasi-particle with effective mass). Introducing a perturbing potential, free electron states will start to hybridize which leads to level splittings. If the perturbation has a lattice periodicity, one can define Brillouin zones in (momentum) $k$-space and can see how hybridization at Brillouin zone boundaries leads to formation of bands (also see other answers about Bloch states). These band splittings also occur in an amorphous/aperiodic solid, but there are no Brillouin zones one could fold these bands back into (due to the aperiodicity of the system in real space, also momentum space lacks periodicity: there is no reciprocal lattice).

*Start with isolated atoms (tight-binding approach). As also stated in previous comments, bring two atoms together, and the first, still discrete level splittings appear; bring many atoms together, and these splittings will be broadened into bands. If these atoms are arranged periodically on a lattice, there will again be periodicity in $k$-space as well, and the band structure can be folded back into a Brillouin zone. Also here, lack of periodicity doesn't mean lack of broadening of states into bands, it only means lack of reciprocal lattice and Brillouin zones.


A nice real-world example of existence of electron bands in an aperiodic solid is amorphous SiO2, i.e. the major component of common glass. It is transparent to visible light, because corresponding photon energies do not suffice to create electron-hole pairs by overcoming the band gap energy of amorphous SiO2. Photons with energies from the ultraviolet spectrum, however, have enough energy to excite electrons from the valence band to the conduction band of glass (i.e. the photons are absorbed), explaining why glass is (mostly) opaque to ultraviolet light. (Here is an abstract of a medical paper, that summarizes how efficient different types of glass are at blocking UV light.)
Regarding a citation about band structure in amorphous solids: the following article discusses the optical band gap dependence of soda lime borosilicate glass (measured band gap $\Rightarrow$ experimental evidence for electronic energy bands) as a function of TiO2 dopant concentration:
Ruengsri, Kaewkhao, Limsuwan, Procedia Engineering 32, 772 (2012) (open access).
Figures 3 and 4 of the following article: Roth, "Tight-Binding Models of Amorphous Systems: Liquid Metals", Phys. Rev. B 7, 4321 (1973) (behind pay wall) show density of states for a (tight-binding) model of an amorphous solid: the broad density of states reveal bands (i.e. there are no individual discrete "spikes" in the density of states indicative of isolated states).
A: As you can see from this paper (just an example among many), it is perfectly possible to speak about band structure even in amorphous or liquid systems. In a way the problem is simply matter of a sloppy terminology: the meaning of bands of energy was the bare fact that only some intervals of energy were allowed in a solid, and the easiest solids to be explored were the crystalline solids where bands are further decomposed into k-dependent eigenvalues. Actually, the correct way of naming things should be: 


*

*bands are the allowed interval of energy (in some cases such intervals may be overlapping;

*the k-structure of a band in a crystal should be referred to as the dispersion of the band


In classical textbooks like Ascroft&Mermin, you'll find the Bloch wavefunctions labeled with the Bloch wavevector and an integer which is the band index.
A: Indeed, the easy way to calculate condensed matter is by using the periodic boundary conditions of crystals. But usually, not much happens with the electronic structure on melting: metals remain metals, the density does not change much. Optical, electrical and thermal properties are generally approximately the same.
The main features of the band structure can often be understood from atomic orbitals and a tight-binding starting point. Generally, melting does not change the local coordination.
A: Energy band functions and Bloch waves only exist in periodic models — after all the Bloch-Floquet transform is just a discrete Fourier transform. So when you break periodicity, then energy bands cease to exist. 
You can still define notions like the density of states in non-periodic models, though. In case when the disorder is weak, all essential properties are modifications of the properties of the corresponding periodic systems. The density of states, for example, is very similar to the periodic model but has exponential tails in the band gaps. 
A: The Bloch part is absolutely true: periodicity is needed in order to apply Bloch's theorem, and then derive the properties we're used to. Even a tiny potential, provided that it is periodic, produces band formation.
But don't get lost in maths, the thing is that there's a bunch of atoms, and Pauli's principle still applies, altough ther's not such periodicity. Electrons cannot have the same quantum numbers, so there must be a level splitting. 
Recall the case of helium. You can regard helium as two hydrogen atoms together. If you don't consider the electrno repulsion, you'll have a ssimilar energy-scheme. However, as soon as you consider interaction between electrons, a level splitting appears. What's more, any perturbation can a level splitting.
Weell, now imagine so many atoms. The splitting is so hard that they're eventually overlapping in bands. You just can't calculate them in the same way as in crystals. But you "can consider them to be bands".
A: Amorphous solids don't have energy bands, mathematically speaking. However, the physics underlying the energy bands may be the same as in a strictly periodic material. For example amorphous silicon and germanium are still semiconductors with similar though not the same, properties. Liquid silicon and germanium are, however, metals. As long as the local coordination is not affected, disorder may not fundamentally change the electronic properties. What is important for the electronic structure is whether electronic orbitals are localized or delocalized. In a simplified model, the Hubbard model, this is determined by the ratio between electron hopping and electron-electron onsite repulsion.
