  Noncollinear magnetic structures and multiple magnetic phase transitions in a
sawtooth lattice antiferromagnet consisting of Cr$^{3+}$ are experimentally
identified in this work, thereby proposing the scenario of magnetism-driven
ferroelectricity in a sawtooth lattice. The title compound, BeCr$_2$O$_4$,
displays three magnetic phase transitions at low temperatures, at
$T_{N1}\approx$ 7.5 K, at $T_{N2}\approx$ 25 K and at $T_{N3}\approx$ 26 K,
revealed through magnetic susceptibility, specific heat and neutron diffraction
in this work. These magnetic phase transitions are found to be influenced by
externally applied magnetic fields. Isothermal magnetization curves at low
temperatures below the magnetic transitions indicate the antiferromagnetic
nature of \bco\ with two spin-flop-like transitions occurring at
$H_{c1}\approx$ 29 kOe and $H_{c2} \approx$ 47 kOe. Our high-resolution X-ray
and neutron diffraction studies, performed on single crystal and powder samples
unambiguously determined the crystal structure as orthorhombic $Pbnm$. By
performing the magnetic superspace group analysis of the neutron diffraction
data at low temperatures, the magnetic structure in the temperature range
$T_{N3,N2} < T < T_{N1}$ is determined to be the polar magnetic space group,
$P21nm.1^{\prime}(00g)0s0s$ with a cycloidal magnetic propagation vector
$\textbf{k}_1$ = (0, 0, 0.090(1)). The magnetic structure in the newly
identified phase below $T_{N1}$, is determined as $P21/b.1^{\prime}[b](00g)00s$
with the magnetic propagation vector $\textbf{k}_2$ = (0, 0, 0.908(1)). The
cycloidal spin structure determined in our work is usually associated with
electric polarization, thereby making \bco\ a promising multiferroic belonging
to the sparsely populated family of sawtooth lattice antiferromagnets.
