It is shown that every almost linear mapping h : $A{ ightarrow}B$ of a unital PoissonC*-algebra A to a unital Poisson C*-algebra B is a Poisson C*-algebra homomorph when $h(2^nuy);=;h(2^nu)h(y)$ or $h(3^nuy);=;h(3^nu)h(y)$ for all $y;in;A$, all unitary elements $u;in;A$ and n = 0, 1, 2,$codts$, and that every almost linear almost multiplicative mapping h : $A{ ightarrow}B$ is a Poisson C*-algebra homomorphism when h(2x) = 2h(x) or h(3x) = 3h(x for all $x;in;A$. Here the numbers 2, 3 depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings. We prove the Cauchy-Rassias stability of Poisson C*-algebra homomorphisms in unital Poisson C*-algebras, and of homomorphisms in Poisson Banach modules over a unital Poisson C*-algebra.