  This is an expository paper. A $1$-cycle in a graph is a set $C$ of edges
such that every vertex is contained in an even number of edges from $C$. E.g.,
a cycle in the sense of graph theory is a $1$-cycle, but not vice versa. It is
easy to check that the sum (modulo $2$) of $1$-cycles is a $1$-cycle. In this
text we study the following problems: to find
  $\bullet$ the number of all 1-cycles in a given graph;
  $\bullet$ a small number of 1-cycles in a given graph such that any 1-cycle
is the sum of some of them.
  We also consider generalizations (of these problems) to graphs with symmetry,
and to $2$-cycles in $2$-dimensional hypergraphs.
