  We study a number of combinatorial and algebraic structures arising from
walks on the two-dimensional integer lattice. To a given step set
$X\subseteq\mathbb Z^2$, there are two naturally associated monoids: $\mathscr
F_X$, the monoid of all $X$-walks/paths; and $\mathscr A_X$, the monoid of all
endpoints of $X$-walks starting from the origin $O$. For each $A\in\mathscr
A_X$, write $\pi_X(A)$ for the number of $X$-walks from $O$ to $A$. Calculating
the numbers $\pi_X(A)$ is a classical problem, leading to Fibonacci, Catalan,
Motzkin, Delannoy and Schroder numbers, among many other well-studied sequences
and arrays. Our main results give relationships between finiteness properties
of the numbers $\pi_X(A)$, geometrical properties of the step set $X$,
algebraic properties of the monoid $\mathscr A_X$, and combinatorial properties
of a certain bi-labelled digraph naturally associated to $X$. There is an
intriguing divergence between the cases of finite and infinite step sets, and
some constructions rely on highly non-trivial properties of real numbers. We
also consider the case of walks constrained to stay within a given region of
the plane. Several examples are considered throughout to highlight the
sometimes-subtle nature of the theoretical results.
