Auxetic solids are materials that possess negative Poisson's ratio. Such materials expand transversely when stretched axially, and contract transversely when axially compressed. Auxetic materials have been initially observed in ferromagnetic films [1], face centered cubic crystals [2], hard cyclic hexamers [3,4], and foams [5-8]. Due to their counter-intuitive properties, auxetic materials have been investigated as smart materials for potential applications including cushion materials [9], stents [10,11], pressure vessels [12], sensors [13], morphing airfoils [14,15], smart folding structures [16], smart metamaterials [17], aeroengine fan blades [18], and vibration dampers [19], to name a few. Arising from their unique properties, the mechanical performance of auxetic solids has been investigated [20-33], including their elastic stabilities [34-36]. Additionally, investigations in the dynamic behavior of auxetic solids and structures have also been performed [37-45]. Although a comprehensive set of available results for the frequencies and mode shapes of free vibration of plates have been provided [46], this information are based on limited range of Poisson's ratio, typically between 0.25 and 0.333. Similarly, established information on the buckling behavior of circular plates is presently confined to conventional Poisson's ratio, especially at $v=0.3$. This paper establishes information on the buckling load and fundamental frequencies of circular plates with Poisson's ratio in the range $-1\u2264v\u22640.5$, so as to allow a comparison to be made between conventional and auxetic circular plates. This paper is therefore divided into two parts, i.e., (a) buckling and (b) vibration of circular auxetic plates. These two seemingly unrelated modes of deformation are consolidated herein on the basis of their overlapping similarities in analysis, particularly in the use of Bessel functions for extracting the critical bucking load and the natural frequencies of circular plates.