We discuss the existence of universal spaces (either in the sense
of embeddings or continuous images) for some classes of scattered
Eberlein compacta. Given a cardinal κ, we consider the
class 𝒮κ of all scattered Eberlein compact spaces K
of weight ≤κ and such that the second Cantor-Bendixson
derivative of K is a singleton. We prove that if κ is an
uncountable cardinal such that κ = 2< κ, then there
exists a space X in 𝒮κ such that every member of
𝒮κ is homeomorphic to a retract of X. We show that it
is consistent that there does not exist a universal space (either
by embeddings or by mappings onto) in 𝒮ω₁. Assuming
that 𝔡= ω₁, we prove that there exists a space
X∈𝒮ω₁, which is universal in the sense of
embeddings. We also show that it is consistent that there exists a
space X∈𝒮ω₁, universal in the sense of embeddings,
but 𝒮ω₁ does not contain an universal element in the
sense of mappings onto.