  Let $X$ be a continuum and let $\varphi:X\rightarrow X$ be a homeomorphism.
To construct a dynamical system $(X,\varphi)$ with interesting dynamical
properties, the continuum $X$ often needs to have some complicated topological
structure. In this paper, we are interested in one such dynamical property:
transitivity. By now, various examples of continua $X$ have been constructed in
such a way that the dynamical system $(X,\varphi)$ is transitive. Mostly, they
are examples of continua that are not path-connected, such as the pseudo-arc or
the pseudo-circle, or they are examples of locally connected continua (and
every locally connected continuum is path-connected), Wazewski's universal
dendrite and the Sierpinski carpet are such examples.
  In this paper, we present an example of a dynamical system $(X,\varphi)$,
where $\varphi$ is a homeomorphism on the continuum $X$ and $X$ is a
path-connected but not locally connected continuum. We construct a transitive
homeomorphism on the Lelek fan. As a by-product, a non-invertible transitive
map on the Lelek fan is also constructed.
