If G is an undirected graph, V(G) is its vertex set and

$A\subseteq V\left(G\right)$, then a median of A is any vertex of G which has the minimal sum of distance from all vertices of A. If each subset of V(G) having three vertices has exactly one median, then G is called a median graph. The properties of median graphs are studied. The interrelation between median graphs and median semilattices is shown; a median semilattice is a meet semilattice

$(X,\le )$ such that every principle ideal

$\{$ $x|x\le a\}$ is a distributive lattice and any three elements have an upper bound whenever each pair of them does. At the end of the paper the concept of a local median is introduced and interrelations between medians and Condorcet vertices are described.