  Recent developments in matrix-product-state (MPS) investigations of many-body
localization (MBL) are reviewed, with a discussion of benefits and limitations
of the method. This approach allows one to explore the physics around the MBL
transition in systems much larger than those accessible to exact
diagonalization. System sizes and length scales that can be controllably
accessed by the MPS approach are comparable to those studied in
state-of-the-art experiments. Results for 1D, quasi-1D, and 2D random systems,
as well as 1D quasi-periodic systems are presented. On time scales explored (up
to $t \approx 300$ in units set by the hopping amplitude), a slow, subdiffusive
transport in a rather broad disorder range on the ergodic side of the MBL
transition is found. For 1D random spin chains, which serve as a "standard
model" of the MBL transition, the MPS study demonstrates a substantial drift of
the critical point $W_c(L)$ with the system size $L$: while for $L \approx 20$
we find $W_c \approx 4$, as also given by exact diagonalization, the MPS
results for $L = 50$--100 provide evidence that the critical disorder
saturates, in the large-$L$ limit, at $W_c \approx 5.5$. For quasi-periodic
systems, these finite-size effects are much weaker, which suggests that they
can be largely attributed to rare events. For quasi-1D ($d\times L$, with $d
\ll L$) and 2D ($L\times L$) random systems, the MPS data demonstrate an
unbounded growth of $W_c$ in the limit of large $d$ and $L$, in agreement with
analytical predictions based on the rare-event avalanche theory.
