  For the perimeter length $L_n$ and the area $A_n$ of the convex hull of the
first $n$ steps of a planar random walk, this thesis study $n \to \infty$ mean
and variance asymptotics and establish distributional limits. The results apply
to random walks both with drift (the mean of random walk increments) and with
no drift under mild moments assumptions on the increments.
  Assuming increments of the random walk have finite second moment and non-zero
mean, Snyder and Steele showed that $n^{-1} L_n$ converges almost surely to a
deterministic limit, and proved an upper bound on the variance Var$[ L_n] =
O(n)$. We show that $n^{-1}$Var$[L_n]$ converges and give a simple expression
for the limit, which is non-zero for walks outside a certain degenerate class.
This answers a question of Snyder and Steele. Furthermore, we prove a central
limit theorem for $L_n$ in the non-degenerate case.
  Then we focus on the perimeter length with no drift and area with both drift
and zero-drift cases. These results complement and contrast with previous work
and establish non-Gaussian distributional limits. We deduce these results from
weak convergence statements for the convex hulls of random walks to scaling
limits defined in terms of convex hulls of certain Brownian motions. We give
bounds that confirm that the limiting variances in our results are non-zero.
