  Let X,Y be two separable Banach or Frechet spaces , and (Tn) , n=1,2,... be a
sequence from linear and continuous operators from X to Y . We say that the
sequence (Tn) , n=1,2,... is universal , if there exists some vector v in X
such that the sequence Tn(v) , n=1,2,... is dense in Y . If X=Y we say that the
sequence (Tn) is hypercyclic .More generally we consider an uncountable subset
A from complex numbers and for every fixed a in A we consider a sequence (Ta,n)
, n=1,2,... from linear and continuous operators from X to Y .The problem of
common universal or hypercyclic vectors is whether the uncountable family of
sequences of operators (Ta,n) , n=1,2,... share a common universal vector for
all a in A .We examine , in this work ,some specific cases of this problem for
translation , differential , and backward shift operators . We study also some
approximating problems about universal Taylor series .
