  We establish non-Hermitian topological mechanics in one dimensional (1D) and
two dimensional (2D) lattices consisting of mass points connected by meta-beams
that lead to odd elasticity. Extended from the "non-Hermitian skin effect" in
1D systems, we demonstrate this effect in 2D lattices in which bulk elastic
waves exponentially localize in both lattice directions. We clarify a proper
definition of Berry phase in non-Hermitian systems, with which we characterize
the lattice topology and show the emergence of topological modes on lattice
boundaries. The eigenfrequencies of topological modes are complex due to the
breaking of $\mathcal{PT}$-symmetry and the excitations could exponentially
grow in time in the damped regime. Besides the bulk modes, additional localized
modes arise in the bulk band and they are easily affected by perturbations.
These distinguishing features may manifest themselves in various active
materials and biological systems.
