  We develop Random Batch Methods for interacting particle systems with large
number of particles. These methods use small but random batches for particle
interactions, thus the computational cost is reduced from $O(N^2)$ per time
step to $O(N)$, for a system with $N$ particles with binary interactions. On
one hand, these methods are efficient Asymptotic-Preserving schemes for the
underlying particle systems, allowing $N$-independent time steps and also
capture, in the $N \to \infty$ limit, the solution of the mean field limit
which are nonlinear Fokker-Planck equations; on the other hand, the stochastic
processes generated by the algorithms can also be regarded as new models for
the underlying problems. For one of the methods, we give a particle number
independent error estimate under some special interactions. Then, we apply
these methods to some representative problems in mathematics, physics, social
and data sciences, including the Dyson Brownian motion from random matrix
theory, Thomson's problem, distribution of wealth, opinion dynamics and
clustering. Numerical results show that the methods can capture both the
transient solutions and the global equilibrium in these problems.

Numerical Analysis;Numerical Analysis;Dynamical Systems;Probability