Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic fieldH (or the corresponding lattice gas model with fugacityz=const. exp(−2mHβ) wherem is the magnetic moment of each spin). We show that, in the limit of an infinite lattice, (i) the free energy per site and the distribution functionsn s (x 1, ...,x s ; β,z) are analytic in the two variables β andH if the reciprocal temperature β>0 and the complex numberH is not a limit point of zeros of the grand partition function ξ, and (ii) the Ursell functionsu s (x 1, ...,x s ; β,z) tend to 0 as Δ s ≡Max i, j |x i −x j | → ∞ if β>0 and ReH≠0; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value λ, then |u s |<C exp [(−2 βm |ReH|+e) Δ s /λ] where e is any small positive number andC is independent of Δ s . One consequence of the result (i) is that a phase transition can occur as β is varied at constantH only ifH is a limit point of zeros of ξ (which can happen only if ReH=0); this supplements Lee and Yang's result that the same condition is necessary for a phase transition whenH is varied at constant β. For a lattice or continuum gas with non-negative interaction potential (corresponding, in the lattice case, to an Ising antiferromagnet), similar results are shown to hold provided β>0 and the complex fugacityz is less than the radius of convergence of the Mayerz expansion; for the continuum gas, however,n s andu s must be replaced by their values integrated over small volumes surrounding each of the pointsx 2, ...,x s . It is shown that the pressurep is analytic in both β andz, if it is analytic inz at fixed β over a suitable range of values of β andz, and further that, except for continuum systems without hard cores,p,n s andu s have convergent Maclaurin expansions in β for small enoughz.