  We introduce new fast canonical local algorithms for discrete and continuous
spin systems. We show that for a broad selection of spin systems they compare
favorably to the known ones except for the Ising +/-1 spins. The new procedures
use discretization scheme and the necessary information have to be stored in
computer memory before the simulation. The models for testing discrete spins
are the Ising +/-1, the general Ising S or Blume-Capel model, the Potts and the
clock models. The continuous spins we examine are the O(N) models, including
the continuous Ising model (N=1), the \phi^4 Ising model (N=1), the XY model
(N=2), the Heisenberg model (N=3), the \phi^4 Heisenberg model (N=3), the O(4)
model with applications to the SU(2) lattice gauge theory, and the general O(N)
vector spins with N\ge5.
