  We study the record statistics of random walks after $n$ steps, $x_0,
x_1,\ldots, x_n$, with arbitrary symmetric and continuous distribution
$p(\eta)$ of the jumps $\eta_i = x_i - x_{i-1}$. We consider the age of the
records, i.e. the time up to which a record survives. Depending on how the age
of the current last record is defined, we propose three distinct sequences of
ages (indexed by $\alpha$ = I, II, III) associated to a given sequence of
records. We then focus on the longest lasting record, which is the longest
element among this sequence of ages. To characterize the statistics of these
longest lasting records, we compute: (i) the probability that the record of the
longest age is broken at step $n$, denoted by $Q^{\alpha}(n)$, which we call
the probability of record breaking and: (ii) the duration of the longest
lasting record, $\ell_{\max}^{\alpha}(n)$. We show that both $Q^{\alpha}(n)$
and the full statistics of $\ell_{\max}^{\alpha}(n)$ are universal, i.e.
independent of the jump distribution $p(\eta)$. We compute exactly the large
$n$ asymptotic behaviors of $Q^{\alpha}(n)$ as well as $\langle
\ell_{\max}^{\alpha}(n)\rangle$ (when it exists) and show that each case gives
rise to a different universal constant associated to random walks (including
L\'evy flights). While two of them appeared before in the excursion theory of
Brownian motion, for which we provide here a simpler derivation, the third case
gives rise to a non-trivial new constant $C^{\rm III} = 0.241749 \ldots$
associated to the records of random walks. Other observables characterizing the
ages of the records, exhibiting an interesting universal behavior, are also
discussed.
