We consider the semilinear heat equation $u_t=\Delta u+u^p$ on
$\mathbb R^N.$ Assuming that $N\ge 3$ and $p$ is greater than the
Sobolev critical exponent $(N+2)/(N-2)$, we examine
entire solutions (classical solutions defined for all $t\in \mathbb R$)
and ancient solutions (classical solutions defined on
$(-\infty,T)$ for some $T<\infty$). We prove a new Liouville-type
theorem saying that if $p$ is greater than the Lepin exponent
$p_L:=1+6/(N-10)$ ($p_L=\infty$ if $N\le 10$), then all positive
bounded radial entire solutions are steady states. The theorem is not
valid without the assumption of radial symmetry; in other ranges
of supercritical $p$ it is known not to be valid even in the class of
radial solutions. Our other results
include classification theorems for nonstationary entire
solutions (when they exist) and ancient solutions, as well as some
applications in the theory of blowup of solutions.