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We show that bosonic and fermionic Gaussian states (also known as``squeezed coherent states’’) can be uniquely characterized by theirlinear complex structure J J which is a linear map on the classical phase space. This extendsconventional Gaussian methods based on covariance matrices and providesa unified framework to treat bosons and fermions simultaneously. PureGaussian states can be identified with the triple (G,\Omega,J) ( G , Ω , J ) of compatible Kähler structures, consisting of a positive definitemetric G G ,a symplectic form \Omega Ω and a linear complex structure J J with J^2=-\mathbb{1} J 2 = − 1 .Mixed Gaussian states can also be identified with such a triple, butwith J^2\neq -\mathbb{1} J 2 ≠ − 1 .We apply these methods to show how computations involving Gaussianstates can be reduced to algebraic operations of these objects, leadingto many known and some unknown identities. We apply these methods to thestudy of (A) entanglement and complexity, (B) dynamics of stablesystems, (C) dynamics of driven systems. From this, we compile acomprehensive list of mathematical structures and formulas to comparebosonic and fermionic Gaussian states side-by-side.