  For a sequence $x \in l_1 \setminus c_{00}$, one can consider the achievement
set $E(x)$ of all subsums of series $\sum_{n=1}^{\infty} x(n)$. It is known
that $E(x)$ is one of the following types of sets:
  * finite union of closed intervals,
  * homeomorphic to the Cantor set,
  * homeomorphic to the set $T$ of subsums of $\sum_{n=1}^{\infty} c(n)$ where
$c(2n-1)=\frac{3}{4^n}$ and $c(2n)=\frac{2}{4^n}$ (Cantorval).
  Based on ideas of Jones and Velleman, and Guthrie and Nymann we describe
families of sequences which contain, according to our knowledge, all known
examples of $x$'s with $E(x)$ being Cantorvals.
