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We find analytical vacuum stability or bounded below conditions for generalscalar potentials of a few fields. After a brief review of copositivity we gobeyond it. We discuss the vacuum stability conditions of the general potentialof two real scalars, without and with the Higgs boson included in thepotential. As further examples, we give explicit vacuum stability conditionsfor the two Higgs doublet model with no explicit CP breaking, and for the$\mathbb{Z}_{3}$ scalar dark matter with an inert doublet and a complexsinglet. We give a short overview of positivity conditions for tensors ofquartic couplings via tensor eigenvalues. A Mathematica notebook with theconditions is included with the source files.

Vacuum Stability of a General Scalar Potential of a Few Fields