  We address the enumeration of walks with small steps confined to a
two-dimensional cone, for example the quarter plane, three-quarter plane or the
slit plane. In the quarter plane case, the solutions for unweighted step-sets
are already well understood, in the sense that it is known precisely for which
cases the generating function is algebraic, D-finite or D-algebraic, and exact
integral expressions are known in all cases. We derive similar results in a
much more general setting: we enumerate walks on an $M$-quadrant cone for any
positive integer $M$, with weighted steps starting at any point. The main
breakthrough in this work is the derivation of an analytic functional equation
which characterises the generating function of these walks, which is analogous
to one now used widely for quarter-plane walks. In the case $M=3$, which
corresponds to walks avoiding a quadrant, we provide exact integral-expression
solutions for walks with weighted small steps which determine the generating
function ${\sf C}(x,y;t)$ counting these walks. Moreover, for each step-set and
starting point of the walk we determine whether the generating function ${\sf
C}(x,y;t)$ is algebraic, D-finite or D-algebraic as a function of $x$ and $y$.
In fact we provide results of this type for any $M$-quadrant cone, showing that
this nature is the same for any odd $M$. For $M$ even we find that the
generating functions counting these walks are D-finite in $x$ and $y$, and
algebraic if and only if the starting point of the walk is on the same axis as
the boundaries of the cone.

Combinatorics;Probability