  We present a brief survey on the modern tensor numerical methods for
multidimensional stationary and time-dependent partial differential equations
(PDEs). The guiding principle of the tensor approach is the rank-structured
separable approximation of multivariate functions and operators represented on
a grid. Recently, the traditional Tucker, canonical, and matrix product states
(tensor train) tensor models have been applied to the grid-based electronic
structure calculations, to parametric PDEs, and to dynamical equations arising
in scientific computing. The essential progress is based on the quantics tensor
approximation method proved to be capable to represent (approximate) function
related $d$-dimensional data arrays of size $N^d$ with log-volume complexity,
$O(d \log N)$. Combined with the traditional numerical schemes, these novel
tools establish a new promising approach for solving multidimensional integral
and differential equations using low-parametric rank-structured tensor formats.
As the main example, we describe the grid-based tensor numerical approach for
solving the 3D nonlinear Hartree-Fock eigenvalue problem, that was the starting
point for the developments of tensor-structured numerical methods for
large-scale computations in solving real-life multidimensional problems. We
also address new results on tensor approximation of the dynamical Fokker-Planck
and master equations in many dimensions up to $d=20$. Numerical tests
demonstrate the benefits of the rank-structured tensor approximation on the
aforementioned examples of multidimensional PDEs. In particular, the use of
grid-based tensor representations in the reduced basis of atomics orbitals
yields an accurate solution of the Hartree-Fock equation on large $N\times N
\times N$ grids with a grid size of up to $N= 10^{5}$.
