Every definition of art has been defeated by an artwork, and usually deliberately. Twentieth-century artists made it a project to locate each proposed boundary and step across it, which means the definitional problem is not a failure of philosophical ingenuity but a feature of the thing being defined.
The oldest account is representation: art imitates reality. It handles portraiture and landscape well, and it collapses on abstract painting, most instrumental music, and architecture, none of which depict anything. Expression theories replace imitation with feeling — art transmits the artist's emotional state to an audience. This covers more, but it fails on work made precisely to be cold and conceptual, and it wrongly implies that a work's value depends on facts about the artist's inner life at the time of making.
Formalism looks instead at arrangement: what makes something art is significant form, the relations between line, colour, mass, or sound. This handles abstraction elegantly. It fails on work whose point is entirely its idea, where the physical object is almost incidental to what is being said.
Duchamp made the difficulty impossible to ignore. A mass-produced urinal, submitted to an exhibition, functioning as art — with nothing about the object distinguishing it from thousands of identical ones. If it is art and they are not, then art-status does not supervene on the object's intrinsic properties at all. Whatever confers it lies in context, intention, or practice.
That pushed theorists toward the institutional theory: something is art when someone acting on behalf of the artworld confers that status on it. This has the considerable merit of describing how the category actually operates, and the considerable cost of appearing to make art whatever the relevant institutions declare, with no independent standard and no way to say they got it wrong. Historical theories offer a variant: something is art if it is intended for regard in a way earlier artworks were regarded, which grounds the practice in its own past rather than in an authority.
The most deflationary response takes its cue from Wittgenstein. Perhaps art is a cluster concept — a family resemblance where members share overlapping features without any single feature being common to all, as with games. On this view, seeking necessary and sufficient conditions was always the wrong project, and the right question is not what art is but what asking that question is for. Notice that in practice the definitional question rarely matters on its own: it usually surfaces because something else is at stake — funding, exhibition, whether a work deserves serious attention. Which suggests the philosophical puzzle may be downstream of practical decisions we would have to make either way.