Proton-proton (pp) collisions have been used extensively as a reference for the study of interactions of larger colliding systems at the LHC. Recent measurements performed in high-multiplicity pp and proton-lead (p-Pb) collisions have shown features that are reminiscent of those observed in lead-lead (Pb-Pb) collisions... In this context, the study of identified particle spectra and yields as a function of multiplicity is a key tool for the understanding of similarities and differences between small and large systems. We report on the production of pions, kaons, protons, $K^{0}_{\rm S}$, $\Lambda$, $\Xi$, $\Omega$ and $K^{*0}$ as a function of multiplicity in pp collisions at $\sqrt{s}=$ 7 TeV measured with the ALICE experiment. The work presented here represents the most comprehensive set of results on identified particle production in pp collisions at the LHC. Spectral shapes, studied both for individual particles and via particle ratios as a function of $p_{\rm T}$, exhibit an evolution with charged particle multiplicity that is similar to the one observed in larger systems. In addition, results on the production of light flavour hadrons in pp collisions at $\sqrt{s}=$ 13 TeV, the highest centre-of-mass energy ever reached in the laboratory, are also presented and compared with previous, lower energy results. (read more)