On the existence of free algebras for the negative theories. We establish conditions on a set $\Sigma$ of negative sentences (in a finitary algebraic language) for which its class of models $M(\Sigma)$ has free algebras (in the classical sense as well as In the "categorical" sense). This is shown to be equivalent to identify the "positive properties" on the absolutely free algebras. The answer depends heavily on the set $\tau$ of operation symbols in the language. For example, if $\tau$ is infinite, then $M(\Sigma)$ always has all free algebras. When $\tau$ is finite, there exists a (positive) sentence $\psi$ (depending only on $\tau$) which is such that $M(\Sigma)$ has free algebras if and only if $\Sigma\cup\{\psi\}$-is constant.