  A class of interacting particle systems on $\mathbb{Z}$, involving
instantaneously annihilating or coalescing nearest neighbour random walks, are
shown to be Pfaffan point processes for all deterministic initial conditions.
As diffusion limits, explicit Pfaffan kernels are derived for a variety of
coalescing and annihilating Brownian systems. For Brownian motions on
$\mathbb{R}$, depending on the initial conditions, the corresponding kernels
are closely related to the bulk and edge scaling limits of the Pfaffan point
process for real eigenvalues for the real Ginibre ensemble of random matrices.
For Brownian motions on $\mathbb{R}_{+}$ with absorbing or reflected boundary
conditions at zero new interesting Pfaffan kernels appear. We illustrate the
utility of the Pfaffan structure by determining the extreme statistics of the
rightmost particle for the purely annihilating Brownian motions, and also
computing the probability of overcrowded regions for all models.
