We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables $\{X_k\}$ of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables $\{X_{ij}\}_{j>i}$ of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions $γ_H$, $γ_M$ and $γ_T$ of unbounded support. The moments of $γ_H$ and $γ_T$ are the sum of volumes of solids related to Eulerian numbers, whereas $γ_M$ has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables $\{X_{ij}\}_{j>i}$ of mean $m$ and finite variance, scaling the eigenvalues by ${n}$ we prove the almost sure, weak convergence of the spectral measures to the atomic measure at $-m$. If $m=0$, and the fourth moment is finite, we prove that the spectral norm of $\mathbf {M}_n$ scaled by $\sqrt{2n\log n}$ converges almost surely to 1.