  We study the distribution of large (and small) values of several families of
$L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider
the Riemann zeta function $\zeta(s)$ in the $t$-aspect, Dirichlet $L$-functions
in the $q$-aspect, and $L$-functions attached to primitive holomorphic cusp
forms of weight $2$ in the level aspect. For each family we show that the
$L$-values can be very well modeled by an adequate random Euler product,
uniformly in a wide range. We also prove new $\Omega$-results for quadratic
Dirichlet $L$-functions (predicted to be best possible by the probabilistic
model) conditionally on GRH, and other results related to large moments of
$\zeta(\sigma+it)$.
