Definition
The no-boundary proposal — formulated by James Hartle and Stephen Hawking in 1983 — is a candidate boundary condition for the wave function of the universe: spacetime, viewed in imaginary time, has no edge or boundary, like the surface of a sphere has no edge in space.
In ordinary real time the universe appears to have a beginning at the Big Bang. In imaginary time, the geometry smoothly closes off — there is no "before," because there is no boundary on which "before" could live.
Why it matters
How it works
In quantum mechanics, the state of a system is described by a wave function obeying the Schrödinger equation. Applying this picture to the universe as a whole was Hugh Everett's and John Wheeler's program, formalized by Bryce DeWitt as the Wheeler-DeWitt equation. The catch: this equation has many solutions, and choosing among them requires a boundary condition — analogous to saying how a violin string is fixed at its ends.
Hartle and Hawking proposed defining the wave function by a Euclidean path integral: sum over all four-dimensional Riemannian geometries that have no boundary at all, except for a final three-dimensional slice representing the present-day spatial universe. There is no "initial" boundary to choose initial conditions on — there is no initial moment in imaginary time.
In ordinary real time this looks bizarre. But the technique is the cosmological analogue of how Hawking radiation was derived: continue to imaginary time, do the calculation in a regime that is smooth and well-defined, then continue back. In real time, the universe appears to emerge from a Big Bang at t = 0; in imaginary time, the geometry is a smooth sphere-like cap with no edge. The most probable histories in this sum are ones that exit imaginary time into a long inflationary phase, producing the smooth, flat, expanding universe we observe — and whose low initial entropy sets the arrow of time.