For a compact metric space ((K, d), \alpha \in(0,1]) and (f \in C(K)), let (p_{x}(f)=) (\sup \left{|f(t)-f(s)| d(t, s)^{x}: t, s \in K\right}). The set (\operatorname{Lip}{x}(K, d)=\left{f \in C(K): p{x}(f)<\infty\right}) with the norm (|f|{x}=|f|{\kappa}+p_{x}(f)) is a Banach function algebra under pointwise multiplication. The subset (\operatorname{lip}{x}(K, d)=\left{f \in \operatorname{Lip}{x}(K, d):|f(t)-f(s)| / d(t, s)^{x} \rightarrow 0\right.) as (d(t, s) \rightarrow 0 ;) is a closed subalgebra of (\operatorname{Lip}{x}(K, d)). For (0<x<\beta, \operatorname{lip}{x}(K, d) ?) (\operatorname{Lip}{\beta}(K, d) \supseteq \operatorname{lip}{\beta}(K, d)), and so they form a one parameter family of algebras ordered by inclusion. Let (A_{x}=\operatorname{Lip}{z}(K, d)) or (\operatorname{lip}{x}(K, d)). For (\alpha, \beta \in(0,1)), the relationship between the ideals of (A_{x}) and (A_{\beta}) is examined, and important inclusions of different such ideals derived. This enables us to establish automatic continuity properties of Lipschitz algebras. A homomorphism (v: A_{x} \rightarrow B, B) a Banach algebra, is said to be eventually continuous if (\exists \beta \geqslant \alpha) such that (v \mid A_{\beta}) is continuous for the (|\left.\cdot\right|{\beta})-norm. First, it is shown that in Lipschitz algebras, when (\alpha \in\left(0, \frac{1}{2}\right)), the eventual continuity is equivalent to nilpotency of the separating ideal in the range algebra. This is used to prove that if (\alpha \in\left(0, \frac{1}{2}\right)), and (v) is eventually continuous, then (v) is continuous on (A{2 x+;}) for all (; \in(0,1-2 \alpha)). It is also shown that in these algebras the prime ideals containing a given prime ideal form a chain. All these results are then used to prove that for (\alpha \in\left(0, \frac{1}{2}\right)) every epimorphism from (A_{x}) is eventually continuous. This research extends work of Bade. Curties, and Laursen, who considered these same questions for (C^{\prime \prime}([0.1]) . \quad) is 1995 Academic Press. Inc.