We start with Universal Taylor series of one variable depending on a parameter. We continue with C∞ 2 π - periodic functions having a Universal Fourier series. This leads to the fact that extendability from one side of the curve of definition is a rare phenomenon in the spaces Cp(∂Ω); p ∈ {∞} ∪{0, 1, ...}. Similar results are obtained in the spaces Ap(Ω) which spaces are studied systematically and lead to the notion of p - continuous analytic capacity.