  We introduce stochastic models of chemotaxis generalizing the deterministic
Keller-Segel model. These models include fluctuations which are important in
systems with small particle numbers or close to a critical point. Following
Dean's approach, we derive the exact kinetic equation satisfied by the density
distribution of cells. In the mean field limit where statistical correlations
between cells are neglected, we recover the Keller-Segel model governing the
smooth density field. We also consider hydrodynamic and kinetic models of
chemotaxis that take into account the inertia of the particles and lead to a
delay in the adjustment of the velocity of cells with the chemotactic gradient.
We make the connection with the Cattaneo model of chemotaxis and the telegraph
equation.
