Woodward-Hoffmann rules
The Woodward-Hoffmann rules devised by Robert Burns Woodward and Roald Hoffmann are a set of rules in organic chemistry predicting the stereochemistry of pericyclic reactions based on orbital symmetry. These include electrocyclic reactions, cycloadditions, and sigmatropic reactions.  Hoffmann was awarded the 1981 Nobel Prize in Chemistry for this work, shared with Kenichi Fukui who developed a similar model, while Woodward had died two years before he could win a second Nobel Prize for Chemistry.

# Electrocyclic reaction
The rules apply to the observed stereospecificity of electrocyclic ring-opening and ring closing reactions at the termini of open chain conjugated polyenes either by application of heat (thermal reactions) or application of light (photochemical reactions). In the original publication in 1965   three rules are stated as:
- In an open-chain system containing 4n-electrons, the orbital symmetry of the highest occupied ground-state orbital is such that a bonding interaction between the termini must involve overlap between orbital envelopes on opposite faces of the system and this can only be achieved in a conrotatory process. An example of such reaction type is the Nazarov cyclization reaction of divinylketones.
- In open systems containing 4n + 2 electrons, terminal bonding interaction within ground-state molecules requires overlap of orbital envelopes on the same face of the system, attainable only by disrotatory displacements
- In a photochemical reaction an electron in the HOMO of the reactant is promoted to an excited state leading to a reversal of terminal symmetry relationships and reversal of stereospecificity.
Organic reactions that obey these rules are said to be symmetry allowed. Reactions that take the opposite course are symmetry forbidden and require a lot more energy to take place if they take place at all.
The rules predict the outcome of several ground-state reactions:
The stated rules are supported by theoretical calculations using the extended Huckel theory. For example,  the activation energy required for thermal ring closing reaction of butadiene can be calculated as a function of the C-C-C bond angles keeping the other variables constant. Angles larger than 117 show a slight preference for a disrotatory reaction but with smaller angles a conrotatory reaction mode is preferred.

# Controversy
It has been stated  that the chemist E.J. Corey feels he is responsible for the ideas that laid the foundation for this research, and that Woodward unfairly neglected to credit him in the discovery. In a 2004 memoir published in the Journal of Organic Chemistry  Corey makes his claim to fame with the single sentence: On May 4, 1964, I suggested to my colleague R. B. Woodward a simple explanation involving the symmetry of the perturbed (HOMO) molecular orbitals for the stereoselective cyclobutene to 1,3-butadiene and 1,3,5-hexatriene to cyclohexadiene conversions that provided the basis for the further development of these ideas into what became known as the Woodward-Hoffmann rules.
In a 2004 rebuttal published in the Angewandte Chemie Roald Hoffmann   denied the claim: he quotes Woodward from a lecture given in 1966 saying: I REMEMBER very clearlyand it still surprises me somewhatthat the crucial flash of enlightenment came to me in algebraic, rather than in pictorial or geometric  form. Out of the blue, it occurred to me that the coefficients of the terminal terms in the mathematical expression representing the highest occupied molecular orbital of butadiene were of opposite sign, while those of the corresponding expression for hexatriene possessed the same sign. From here it was but a short step to the geometric, and more obviously chemically relevant, view that in the internal cyclisation of a diene, the top face of one terminal atom should attack the bottom face of the other, while in the triene case, the formation of a new bond should involve the top (or pari passu, the bottom) faces of both terminal atoms.
In addition, Hoffmann points out that in 2 publications from 1963  and 1965  Corey described a total synthesis of the compound dihydrocostunolide and although in it an electrocyclic reaction is described Corey has nothing to offer with respect to explaining its stereospecifity.
This photochemical reaction involving 4*1+2 electrons is now recognized as conrotatory.