Fra\“iss\’e sequences and their limits are universal constructions whose impact on functional analysis and Banach space theory is not yet well appreciated. Our rather pedestrian approach is aimed at the construction and study of two concrete examples: the $p$-Gurariy space, namely the only separable $p$-Banach space of almost universal disposition, and the $p$-Kadec space, a separable $p$-Banach space of almost universal complemented disposition with a 1-FDD. The chapter emphasises that these spaces correspond to the same object, but in different categories.