  The statistical rank tests play important roles in univariate non-parametric
data analysis. If one attempts to generalize the rank tests to a multivariate
case, the problem of defining a multivariate order will occur. It is not clear
how to define a multivariate order or statistical rank in a meaningful way. One
approach to overcome this problem is to use the notion of data depth which
measures the centrality of a point with respect to a given data set. In other
words, a data depth can be applied to indicate how deep a point is located with
respect to a given data set. Using data depth, a multivariate order can be
defined by ordering the data points according to their depth values. Various
notions of data depth have been introduced over the last decades. In this
thesis, we discuss three depth functions: two well-known depth functions
halfspace depth and simplicial depth, and one recently defined depth function
named as $\beta$-skeleton depth, $\beta\geq 1$. The $\beta$-skeleton depth is
equivalent to the previously defined spherical depth and lens depth when
$\beta=1$ and $\beta=2$, respectively. Our main focus in this thesis is to
explore the geometric and algorithmic aspects of $\beta$-skeleton depth.
