Consider the Cauchy problem in

${\mathbb{R}}^{N}$ for the equation

${u}_{t}={\Delta}u+{u}^{p}$, where

$p>1$ and

$u\ge 0$. In 1966,

*H. Fujita* [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966;

Zbl 0163.34002)] showed that this problem does not have global nontrivial solutions if

$p<{p}_{c}:=1+2/N$ whereas both global and non-global positive solutions exist if

$p>{p}_{c}$. The exponent

${p}_{c}$ is called Fujita’s critical exponent. The authors discuss various Fujita-type results which have appeared in the literature since 1990. These results include degenerate equations, problems in unbounded domains and on manifolds, problems with inhomogeneous boundary conditions, cooperative systems of equations. Moreover, the paper contains a section with open problems.