  Achieving the highest possible precision for theoretical predictions at the
present and future high-energy lepton and hadron colliders requires a precise
determination of fragmentation functions (FFs) of light and heavy charged
hadrons from a global QCD analysis with great accuracy. We describe a
simultaneous determination of unpolarized FFs of charged pions, charged kaons
and protons/antiprotons from single-inclusive hadron production in
electron-positron annihilation (SIA) data at next-to-leading order and
next-to-next-to-leading order accuracy in perturbative QCD. A new set of FFs,
called {\tt SGKS20}, is presented. We include data for identified light charged
hadrons ($\pi^\pm, K^\pm$ and $p/\bar{p}$) as well as for unidentified light
charged hadrons, $h^\pm$ and show that these data have a significant impact on
both size and uncertainties of the fragmentation functions. We examine the
inclusion of higher-order perturbative QCD corrections and finite-mass effects.
We compare the new {\tt SGKS20} FFs with other recent FFs available in the
literature and find in general reasonable agreement, but also important
differences for some parton species. We show that theoretical predictions
obtained from our new FFs are in very good agreement with the analyzed SIA
data, especially at small values of $z$. The {\tt SGKS20} FF sets presented in
this work are available via the {\tt LHAPDF} interface.
