  In this paper we present a method to obtain Banach spaces of universal and
almost-universal disposition with respect to a given class $\mathfrak M$ of
normed spaces. The method produces, among other, the Gurari\u{\i} space
$\mathcal G$ (the only separable Banach space of almost-universal disposition
with respect to the class $\mathfrak F$ of finite dimensional spaces), or the
Kubis space $\mathcal K$ (under {\sf CH}, the only Banach space with the
density character the continuum which is of universal disposition with respect
to the class $\mathfrak S$ of separable spaces). We moreover show that
$\mathcal K$ is not isomorphic to a subspace of any $C(K)$-space -- which
provides a partial answer to the injective space problem-- and that --under
{\sf CH}-- it is isomorphic to an ultrapower of the Gurari\u{\i} space.
  We study further properties of spaces of universal disposition: separable
injectivity, partially automorphic character and uniqueness properties.
