In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows. We prove that if
X
X
is a metric space and
A
⊂
X
A\subset X
satisfies the condition
Nagata
(
n
,
c
)
\hspace{0.1em}\text{Nagata}\hspace{0.1em}\left(n,c)
, then any 1-Lipschitz map
f
:
A
→
Y
f:A\to Y
to a Banach space
Y
Y
admits a Lipschitz extension
F
:
X
→
Y
F:X\to Y
whose Lipschitz constant is at most
1,000
⋅
(
c
+
1
)
⋅
log
2
(
n
+
2
)
\hspace{0.1em}\text{1,000}\hspace{0.1em}\cdot \left(c+1)\cdot {\log }_{2}\left(n+2)
. By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if
A
⊂
X
A\subset X
consists of
n
n
points, then Lipschitz extensions as above exist with a Lipschitz constant of at most
600
⋅
log
n
⋅
(
log
log
n
)
−
1
600\cdot \log n\cdot {\left(\log \log n)}^{-1}
.