  We introduce a new class of asymmetric random walks on the one-dimensional
infinite lattice. In this walk the direction of the jumps (positive or
negative) is determined by a discrete-time renewal process which is independent
of the jumps. We call this discrete-time counting process the `it generator
process' of the walk. We refer the so defined walk to as `Asymmetric
Discrete-Time Random Walk' (ADTRW). We highlight connections of the
waiting-time density generating functions with Bell polynomials. We derive the
discrete-time renewal equations governing the time-evolution of the ADTRW and
analyze recurrent/transient features of simple ADTRWs (walks with unit jumps in
both directions). We explore the connections of the recurrence/transience with
the bias: Transient simple ADTRWs are biased and vice verse. Recurrent simple
ADTRWs are either unbiased in the large time limit or `strictly unbiased' at
all times with symmetric Bernoulli generator process. In this analysis we
highlight the connections of bias and light-tailed/fat-tailed features of the
waiting time density in the generator process. As a prototypical example with
fat-tailed feature we consider the ADTRW with Sibuya distributed waiting times.
We also introduce time-changed versions: We subordinate the ADTRW to a
continuous-time renewal process which is independent from the generator process
and the jumps to define the new class of `Asymmetric Continuous Time Random
Walk' (ACTRW). This new class - apart of some special cases - is not a
Montroll--Weiss continuous-time random walk (CTRW). ADTRW and ACTRW models may
open large interdisciplinary fields in anomalous transport, birth-death models
and others.
