  P\'osa's theorem states that any graph $G$ whose degree sequence $d_1 \le
\ldots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle.
This degree condition is best possible. We show that a similar result holds for
suitable subgraphs $G$ of random graphs, i.e. we prove a `resilience version'
of P\'osa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered
increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$,
then $G$ has a Hamilton cycle. This is essentially best possible and
strengthens a resilience version of Dirac's theorem obtained by Lee and
Sudakov.
  Chv\'atal's theorem generalises P\'osa's theorem and characterises all degree
sequences which ensure the existence of a Hamilton cycle. We show that a
natural guess for a resilience version of Chv\'atal's theorem fails to be true.
We formulate a conjecture which would repair this guess, and show that the
corresponding degree conditions ensure the existence of a perfect matching in
any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an
asymptotic characterisation of all degree sequences which resiliently guarantee
the existence of a perfect matching.
