  We consider the P\'olya random walk in $\mathbb{Z}^2$. The paper establishes
a number of results for the distributions and expectations of the number of
usual (undirected) and specifically defined in the paper up- and down-directed
state-crossings and different sets of states crossings. One of the most
important results of this paper is that the expected number of undirected
state-crossings $\mathbf{n}$ is equal to 1 for any state
$\mathbf{n}\in\mathbb{Z}^2\setminus\{\mathbf{0}\}$. As well, the results of the
paper are extended to $d$-dimensional random walks, $d\geq2$, in bounded areas.
