The notion of a quasiuniform fat point subscheme
Z
⊂
P
2
Z\subset \mathbf P^2
is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal
I
I
defining
Z
Z
are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the
m
m
th symbolic power
I
(
m
;
n
)
I(m;n)
of an ideal defining
n
n
general points of
P
2
\mathbf P^2
when both
m
m
and
n
n
are large (in particular, for infinitely many
m
m
for each of infinitely many
n
n
, and for infinitely many
n
n
for every
m
>
2
m>2
). Resolutions in other cases, such as “fat points with tails”, are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field
k
k
. As an incidental result, a bound for the regularity of
I
(
m
;
n
)
I(m;n)
is given which is often a significant improvement on previously known bounds.