Mass
Mass is the property by which a body resists being accelerated and by which it gravitates. Those are two distinct descriptions — inertial mass, defined by Newton's second law, and gravitational mass, defined by the law of gravitation — and the fact that they agree is a measured result rather than a definition.
Newton took the equality for granted; Loránd Eötvös tested it around 1900 with a torsion balance, and modern experiments, including the MICROSCOPE satellite mission, have confirmed it to roughly one part in 1015. Einstein made the equality a postulate, the equivalence principle, and built general relativity on it. Special relativity added a second relation: the energy content of a body at rest is E = mc2. In modern usage the mass of a particle means its invariant rest mass; the older notion of a "relativistic mass" increasing with speed has largely been abandoned in the professional literature in favour of speaking about relativistic momentum and energy, though it survives in textbooks.
What mass actually is remains open even on the mainstream account. The Higgs mechanism, confirmed in its broad outline by the 2012 discovery of the Higgs boson, supplies masses to the elementary particles, but the values are free parameters fitted to experiment rather than predicted. And the Higgs accounts for only a small fraction of the mass of ordinary matter: the great majority of a proton's mass is the binding energy of the field inside it. Nothing in the Standard Model explains why inertia exists at all.
On this wiki
Three arguments about mass run through the literature collected here.
Does mass increase with speed? Several researchers hold that it does not. Clarence L Dulaney's Mass Variation With Speed (1999) attacks the derivation itself, arguing that the spherical-wave equation from which the Lorentz transformations are conventionally obtained is written down incorrectly, and that every standard derivation of mass variation inherits the error; he returned to the question with Georg Galeczki, Cynthia Kolb Whitney and Franco Selleri in "General Considerations about Mass Variation" (2000). Curtis E Renshaw argued the same conclusion in "The Untenable Nature of Relativistic Mass Increase" (1999). Autodynamics, developed by Ricardo L Carezani, goes further and predicts mass decrease with speed rather than increase.
Is mass converted into energy? James Carter's E = mc2 Fallacies and the Non-Conversion of Mass to Energy (1997) argues that the usual reading of E = mc2 depends on the arbitrary assumption that the photon is massless; on a classical reading of the same experiments the photon carries mass, and mass and energy remain separate and distinct parameters, neither ever converted into the other. Hartwig Wolfgang Thim's The Long History of the Mass-Energy Relation (2006) traces the relation's pre-Einstein ancestry. The related question of whether energy is a substance at all is taken up on the Energy page.
Where does inertial mass come from? This is the most productive thread. David L Bergman's Origin of Inertial Mass (1999) argues that inertia is not an intrinsic property of matter: the inertial reaction force is a real, self-induced electrodynamic force on the circulating charge of his spinning ring model, which makes inertial mass a derived concept and force the fundamental one. Thomas G Barnes reached the same conclusion earlier in Electric Explanation Of Inertial Mass (1983), deriving the inertial reaction as a magnetically induced electric force on the charges making up an accelerated body. Charles William Lucas's The Electrodynamic Origin of the Force of Inertia (2007) derives both inertial and gravitational mass from the same universal electrodynamic force and obtains their equality as a result rather than a postulate — with the striking corollary that both should decay slowly over time. Roger A Rydin's A Philosophic Discussion of the Concept of Mass and the Principle of Isolation (2013) works through the mass defect, apparent mass increase and cosmological mass questions in terms of Lucas's nuclear model.
A rival answer places the origin of mass outside the body. Andre K T Assis's Relational Mechanics and his paper with Jorge Guala-Valverde, Mass in Relational Mechanics (2000), make inertia a gravitational interaction with the distant matter of the universe: doubling the average gravitational mass density of the distant galaxies would halve the local acceleration of free fall. Peter Graneau argues the same Machian case in Mach's Principle & Nonlocal Mass Interactions (2009), a condensed version of a chapter of In the Grip of the Distant Universe: The Science of Inertia written with Neal Graneau. See Category:Mach's Principle for that literature.
Other approaches in the archive make mass structural or medium-borne: Francis Viren Fernandes's Defining Inertial & Gravitational Mass (2009) identifies inertial mass with the photon-torus fibre and gravitational mass with the aether it contains; Robert F Beck's A Heuristic Paper on the Nature of Mass (2007) makes mass depend fundamentally on spin; and Henrik Vilhelm Broberg's Mass, Energy, Space (1991) treats mass, energy and space as a single subject.