  We review recent results and present new ones on a deterministic
follow-the-leader particle approximation of first and second order models for
traffic flow and pedestrian movements. We start by constructing the particle
scheme for the first order Lighthill-Whitham-Richards (LWR) model for traffic
flow. The approximation is performed by a set of ODEs following the position of
discretised vehicles seen as moving particles. The convergence of the scheme in
the many particle limit towards the unique entropy solution of the LWR equation
is proven in the case of the Cauchy problem on the real line. We then extend
our approach to the Initial-Boundary Value Problem (IBVP) with time-varying
Dirichlet data on a bounded interval. In this case we prove that our scheme is
convergent strongly in $L^1$ up to a subsequence. We then review extensions of
this approach to the Hughes model for pedestrian movements and to the second
order Aw-Rascle-Zhang (ARZ) model for vehicular traffic. Finally, we complement
our results with numerical simulations. In particular, the simulations
performed on the IBVP and the ARZ model suggest the consistency of the
corresponding schemes, which is easy to prove rigorously in some simple cases.

Numerical Analysis;Analysis of PDEs