Summary: An arcwise connected compact metric space \( X\) is called a quasi-graph if there is a positive integer \( N\) with the following property: for every arcwise connected subset \( Y\) of \( X\), the space \( \overline {Y}-Y\) has at most \( N\) arcwise connected components. If a quasi-graph \( X\) contains no Jordan curve, then \( X\) is called a quasi-tree. The structures of quasi-graphs and the dynamics of quasi-graph maps are investigated in this paper. More precisely, the structures of quasi-graphs are explicitly described; some criteria for \( \omega \)-limit points of quasi-graph maps are obtained; for every quasi-graph map \( f\), it is shown that the pseudo-closure of \( R(f)\) in the sense of arcwise connectivity is contained in \( \omega (f)\); it is shown that \( \overline {P(f)}=\overline {R(f)}\) for every quasi-tree map \( f\). Here \( P(f)\), \( R(f)\) and \( \omega (f)\) are the periodic point set, the recurrent point set and the \( \omega \)-limit set of \( f\), respectively. These extend some well-known results for interval dynamics.