  Prethermalization refers to the transient phenomenon where a system
thermalizes according to a Hamiltonian that is not the generator of its
evolution. We provide here a rigorous framework for quantum spin systems where
prethermalization is exhibited for very long times. First, we consider quantum
spin systems under periodic driving at high frequency $\nu$. We prove that up
to a quasi-exponential time $\tau_* \sim e^{c \frac{\nu}{\log^3 \nu}}$, the
system barely absorbs energy. Instead, there is an effective local Hamiltonian
$\hat D$ that governs the time evolution up to $\tau_*$, and hence this
effective Hamiltonian is a conserved quantity up to $\tau_*$. Next, we consider
systems without driving, but with a separation of energy scales in the
Hamiltonian. A prime example is the Fermi-Hubbard model where the interaction
$U$ is much larger than the hopping $J$. Also here we prove the emergence of an
effective conserved quantity, different from the Hamiltonian, up to a time
$\tau_*$ that is (almost) exponential in $U/J$.
