In this paper we survey some recent results on parabolic equations on curved squeezed domains. More specifically, consider the family of semilinear Neumann boundary value problems $$ \alignedat2 u_t &= \Delta u + f(u), &\quad&t> 0,\ x\in \Omega_\varepsilon, \\ \partial_{\nu_\varepsilon}u&= 0, &\quad& t> 0,\ x\in \partial \Omega_\varepsilon \endaligned \leqno{(\text{\rm E}_\varepsilon)} $$ where, for $\varepsilon> 0$ small, the set $\Omega_\varepsilon$ is a thin domain in $\mathbb R^\ell$, possibly with holes, which collapses, as $\varepsilon\to0^+$, onto a (curved) $k$-dimensional submanifold $\mathcal M$ of $\mathbb R^\ell$. If $f$ is dissipative, then equation (E$_\varepsilon$) has a global attractor ${\mathcal A}_\varepsilon$. We identify a "limit" equation for the family (E$_\varepsilon$), establish an upper semicontinuity result for the family ${\mathcal A}_\varepsilon$ and prove an inertial manifold theorem in case $\mathcal M$ is a $k$-sphere.