In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat Kähler metrics on Calabi-Yau manifolds along a smoothing is established, which can be of independent interests.is an extremal transition among Calabi-Yau manifolds, then there exists a family of Ricci-flat Kähler metricsḡ s , s ∈ (0, 1), on M, and a family of Ricci-flat Kähler metricsg t on M t satisfying that {(M,ḡ s )} and {(M t ,g t )} converge to a single compact metric space (X, d X ) in the Gromov-Hausdorff topology,ii) IfM 1 → M 0 M 2 is a flop between two Calabi-Yau manifolds, then there are families of Ricci-flat Kähler metricsḡ i,s , s ∈ (0, 1) onM i (i = 1, 2) such that for a single compact metric space (X, d X ).Furthermore, in both cases X is homeomorphic to M 0 and d X is induced by a Ricci-flat Kähler metric on M 0 \S. In the present paper, we shall prove i) and ii) of the above version of Candelas and de la Ossa's conjecture.Let M 0 be a projective normal Cohen-Macaulay n-dimensional variety with singular set S, and let K M 0 be the canonical sheaf of M 0 ([33]). In this paper, all varieties are assumed to be Cohen-Macaulay. We call M 0 Gorenstein if K M 0 is a rank one locally free sheaf. Assume that M 0 has only canonical singularities, i.e., M 0 is Gorenstein, and for any resolution (M,π),where E are effective exceptional divisors. Consider a resolution (M ,π) of M 0 . If α is an ample class in the Picard group of M 0 ,π * α belongs the boundary of the Kähler cone of M . A resolution (M ,π) of M 0 is called a crepant resolution if KM =π * K M 0 and is called a small resolution if the exceptional subvarietyπ −1 (S) satisfies dim Cπ −1 (S) ≤ n − 2. It is obvious that (M ,π) is crepant if it is a small resolution. If M 0 admits a smoothing (M, π) over a unit disc ∆ ⊂ C with an ample line bundle L on M, then there is an embedding M ֒→ CP N × ∆ such that L m = O ∆ (1)| M for some m ≥ 1, π is a proper surjection given by the restriction of the projection from CP N × ∆ to ∆, and the rank of π * is 1 on M\S. This implies that M t , t ∈ ∆\{0}, have the same underlying differential manifoldM . Moreover, if L is a line bundle on M such that the restriction of L on M 0 is ample, then by Proposition 1.41 in [38] L is ample on π −1 (∆ ′ ) where ∆ ′ ⊆ ∆ is a neighborhood of 0.A Calabi-Yau variety is a simply connected projective normal variety M 0 with trivial canonical sheaf K M 0 ∼ = O M 0 and only canonical singularities. If a Calabi-Yau variety M 0
