We study the random conductance model on the lattice Z d , i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension d ≥ 3 quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed t − 1 5 +ε for d ≥ 4 and t − 1 10 +ε for d = 3. Additionally, in the uniformly elliptic case in low dimensions d = 2, 3 we improve the rate in a quantitative Berry-Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for d ≥ 3 we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.1.1. The model. Let d ≥ 2. We study the nearest-neighbour random conductance model on the d-dimensional Euclidean lattice (Z d , E d ), where E d := {e = {x, x ± e i } : x ∈ Z d , i = 1, . . . , d } denotes the set of non-oriented nearest neighbour edges and {e 1 , . . . , e d } the canonical basis in R d . We endow the graph with positive random weights, which we describe by a family ω = {ω(e), e ∈ E d } ∈ Ω := (0, ∞) E d . We refer to ω(e) as the conductance of an edge e ∈ E d . To simplify notation, for any x, y ∈ Z d , we set ω(x, y) = ω(y, x) := ω({x, y}), ∀ {x, y} ∈ E d , ω(x, y) := 0, ∀ {x, y} ∈ E d , and define the matrix field ω : Z d → R d×d by ω(x) = diag(ω(x, x + e 1 ), . . . , ω(x, x + e d )). 2 1 5 if d = 2, c (t + 1) − 1 5 if d ≥ 3.5