For a symmetric kernel
k
:
X
×
X
→
R
∪
{
+
∞
}
k:X\times X \rightarrow \mathbb {R}\cup \{+\infty \}
on a locally compact metric space
X
X
, we investigate the asymptotic behavior of greedy
k
k
-energy points
{
a
i
}
1
∞
\{a_{i}\}_{1}^{\infty }
for a compact subset
A
⊂
X
A\subset X
that are defined inductively by selecting
a
1
∈
A
a_{1}\in A
arbitrarily and
a
n
+
1
a_{n+1}
so that
∑
i
=
1
n
k
(
a
n
+
1
,
a
i
)
=
inf
x
∈
A
∑
i
=
1
n
k
(
x
,
a
i
)
\sum _{i=1}^{n}k(a_{n+1},a_{i})=\inf _{x\in A}\sum _{i=1}^{n}k(x,a_{i})
. We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy
∑
i
≠
j
N
k
(
a
i
,
a
j
)
\sum _{i\neq j}^{N}k(a_{i},a_{j})
as
N
→
∞
N\rightarrow \infty
that is asymptotically the same as
E
(
A
,
N
)
:=
min
{
∑
i
≠
j
k
(
x
i
,
x
j
)
:
x
1
,
…
,
x
N
∈
A
}
\mathcal {E}(A,N):=\min \{\sum _{i\neq j}k(x_{i},x_{j}):x_{1},\ldots ,x_{N}\in A\}
), and have asymptotic distribution equal to the equilibrium measure for
A
A
. For the case of Riesz kernels
k
s
(
x
,
y
)
:=
|
x
−
y
|
−
s
k_{s}(x,y):=|x-y|^{-s}
,
s
>
0
s>0
, we show that if
A
A
is a rectifiable Jordan arc or closed curve in
R
p
\mathbb {R}^{p}
and
s
>
1
s>1
, then greedy
k
s
k_{s}
-energy points are not asymptotically energy minimizing, in contrast to the case
s
>
1
s>1
. (In fact, we show that no sequence of points can be asymptotically energy minimizing for
s
>
1
s>1
.) Additional results are obtained for greedy
k
s
k_{s}
-energy points on a sphere, for greedy best-packing points (the case
s
=
∞
s=\infty
), and for weighted Riesz kernels. For greedy best-packing points we provide a simple counterexample to a conjecture attributed to L. Bos.