The procedures of fabrication and testing of auxetic foams with closed cells based on foaming a liquid substance and by joining microspheres are discussed. Physically , to obtain an auxetic structure, bending rigidity of elastic rods, plates and shells should strongly depend on the initial curvature. The cells of small size are found mostly to hold their original shape. Large ones show relatively low rigidity, and would get deformed similarly to thin-walled shells when compressed with a possibility of losing stability. Thus, the volumetric compression of a foamed material is mainly realized at the expense of decreased free volume of large cells. Separation of cells according to deformation levels is found to cause auxetic elastic behavior in converted closed cells foams. Technologically, to obtain this auxetics we proposed a two-stage process. It includes the formation of concave cell structure by a permanent volumetric compression of the initial material just after foaming in the solidification state under he action of a liquid or gas. High plasticity of foam materials in this stage allow s us to obtain the re-entrant structure of cells. To obtain a material with non-convex cells we used mostly a gas or liquid under pressure as a forming instrument. After cooling the foam material shows the property of elastic (reversible) deformation. I he homogeneity and isotropy of Poisson's ratio of obtained auxetics are caused by a uniform distribution of the gas or liquid pressure on the sample surface. Some problems of Poisson's ratio minimization for foam materials we have solved by the finite element analysis.