  A quadratic form over a non-archimedian local field of characteristic zero
$F$ is called universal if it is integral and it represents all non-zero
integers of $F$. Xu Fei and Zhang Yang determined all universal quadratic forms
in the case when $F$ is non-dyadic. In the more complicated dyadic case, when
$F$ is a finite extension of $\mathbb Q_2$, they solved the same problem only
in the ternary case. In our paper we solve this problem in the general case.
Our result is given in terms of BONGs (bases of norm generators) but in section
3 of the paper we translate (without a proof) our result in terms of the more
traditional Jordan splittings.
  In the last section we give some results on $n$-universality. We show that it
can be reduced to the cases $n\leq 4$ and we give explicit necessary conditions
for $n$-universality in the case when $n\geq 3$, $n$ odd. (A quadratic form is
called $n$-maximal if it is integral and it represents all non-degenerate
integral quadratic forms of rank $n$.)
