  We develop a powerful new analytic method to construct complete non-compact
G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the
compact exceptional Lie group G2. Our construction starts with a complete
non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M
over B satisfying a necessary topological condition. Our method then produces a
1-parameter family of circle-invariant complete G2-metrics on M that collapses
to the original Calabi-Yau metric on the base B as the parameter converges to
0. The G2-metrics we construct have controlled asymptotic geometry at infinity,
so-called asymptotically locally conical (ALC) metrics, and are the natural
higher-dimensional analogues of the ALF metrics that are well known in
4-dimensional hyperk\"ahler geometry. We give two illustrations of the strength
of our method. Firstly we use it to construct infinitely many diffeomorphism
types of complete non-compact simply connected G2-manifolds; previously only a
handful of such diffeomorphism types was known. Secondly we use it to prove the
existence of continuous families of complete non-compact G2-metrics of
arbitrarily high dimension; previously only rigid or 1-parameter families of
complete non-compact G2-metrics were known.
