  We report a comprehensive study of single crystals of the noncentrosymmetric
rare-earth compound EuPtSi$_{3}$ grown by the optical floating-zone technique.
Measurements of the magnetization, ac susceptibility, and specific heat
consistently establish antiferromagnetic order of localized Eu$^{2+}$ moments
below the N\'{e}el temperature $T_{\mathrm{N}} = 17~\mathrm{K}$, followed by a
second magnetic transition at $T_{\mathrm{N1}} = 16~\mathrm{K}$. For a magnetic
field along the easy $[001]$ axis, the magnetic phase diagram is composed of
these two phases. For fields applied in the magnetically hard basal plane, two
additional phases emerge under magnetic field, where the in-plane anisotropy is
weak with $[100]$ being the hardest axis. At the phase transitions, the
magnetic properties exhibit hysteresis and discrepancies between differential
and ac susceptibility, suggesting slow reorientation processes of mesoscale
magnetic textures. Consistently, powder and single-crystal neutron diffraction
in zero field identify magnetic textures that are modulated on a length scale
of the order of $100~\textrm{\r{A}}$, most likely in the form of N\'{e}el-type
antiferromagnetic cycloids.
