The discrete-time, variable tolerance ℌ_∞ filtering problem is presented. The variable tolerance is specified at every time sample and guarantees a feasible tolerance value without an iterative design process. The variable tolerance ℌ_∞ filtering problem generalizes the standard constant tolerance ℌ_∞ filtering problem and, as a result, leads to a straightforward procedure for approximating the optimal value, γ_∞, of the constant tolerance problem. Furthermore, the more general setting of the variable tolerance problem admits an improved ℌ_∞ filter with a smaller 𝓛₂ gain than the constant tolerance ℌ_∞ filter. Applications of a variable tolerance to the ℌ_∞ control problem are discussed as well.