  We introduced simple microscopic non-Markovian walk models which describe
underlying mechanism of anomalous diffusions. In the models, we considered the
competitions between randomness and memory effects of previous history by
introducing the probability parameters. The memory effects were considered in
two aspects, one is the perfect memory of whole history and the other is the
latest memory improved with time. In the perfect memory model superdiffusion
was induced with the relation the Hurst exponent $H$ to the controlling
parameter $p$ as $H=p$ for $p>1/2$. While in the latest memory enhancement
models, anomalous diffusions involving both superdiffusion and subdiffusion
were induced with the relations $H=(1+\alpha)/2$ and $H=(1-\alpha)/2$ for $0
\le \alpha \le 1$ where $\alpha$ is the parameter controlling the degree of the
latest memory enhancement. Also we found that although the latest memory was
only considered, the memory improved with time results in the long-range
correlations between steps and the correlations increases as time goes. Thus we
suggest the memory enhancement as a novel key origin describing anomalous
diffusions.
