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Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${\left(H_I^i\left(M\right)\right)}_{𝔪}$ is a minimax $R_{𝔪}$-module for all $𝔪\in \mathrm{Max}\left(R\right)$ and for all $i<n$, then the set ${\mathrm{Ass}}_R\left(H_I^n\left(M\right)\right)$ is finite. Also, if $H_I^i\left(M\right)$ is minimax for...