We discuss in two relevant case-studies the rigorous derivation via Gamma-convergence of asymptotic energies accounting for singularities in elastic materials from non-local models (convolution-type integral functionals).In the first part, we are concerned with free-discontinuity functionals à la Griffith, coupling a linearly elastic behavior in the uncracked part of a reference configuration with an energy concentrated on lower-dimensional submanifolds accounting for crack formation. The models we propose feature a nonlocal linearly elastic energy coupled with a truncating potential accounting for the breaking of elastic bonds. Their asymptotic behavior in suitable spaces of weakly differentiable functions with surface discontinuities is analyzed and further research directions are proposed. In the second part, a strain-gradient theory for plasticity is derived as a limit of discrete dislocation fractional energies, avoiding the excision of a so-called core-radius from the reference configuration. Away from dislocations, the stored elastic energy is given in terms of a fractional gradient of order 1-alpha of the strain, extended to the case of incompatible strain fields via the usage of Riesz-type singular convolution operators. As alpha goes to 0, we show that a suitable rescaling of the energies converge to a macroscopic strain-gradient model featuring a positively 1-homogeneous plastic energy on the dislocation density, which coincides with the one recovered by Garroni, Leoni, and Ponsiglione (2010) through a core-radius approach. From joint works with Roberta Marziani (L'Aquila), Stefano Almi (Napoli), Maicol Caponi (Napoli) and Manuel Friedrich (Erlangen).