  We establish both the $\limsup$ and the $\liminf$ law of the iterated
logarithm (LIL), for the capacity of the range of a simple random walk in any
dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL
at dimension $d$ matches that for the volume of the random walk range in
dimension $d-2$, somewhat surprisingly this correspondence breaks down for the
capacity of the range at $d=3$. We further establish such LIL for the Brownian
capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate
deviations bounds for the capacity of the range of a $4$-dimensional simple
random walk.
