Let \ $X$\ \ be a Banach space, and $L(X)$\ be the space of bounded linear
operators from $X$\ \ into $X.$\ $B_{1}[X]$\ denotes the closed unit ball of 
$X,$\ and $S_{1}[X]$\ is unit sphere of $X.$\ An element $T\in S_{1}[L(X)]$\
is called extreme operator if there is no $A\in L(X)$\ such that $\left\Vert
T\pm A\right\Vert \leq 1.$\ The set of extreme points of $S_{1}[X]$\ will be
denoted by $ext(S_{1}[X]).$\ $T$\ $\in S_{1}[L(X)]$\ is called nice if $%
T^{\ast }(ext(S_{1}[L(X^{\ast })])\subseteq ext(S_{1}[L(X^{\ast })])$. The
object of this paper is to give simpler proofs of old results and present
new results on extreme operators of $S_{1}[L(\ell ^{p})].$ We introduce the
concept of k-extreme points. \ Further, we characterize the nice operators
on most of the classical function and sequence spaces. Nice compact
operators on $\ell ^{p}-$spaces are characterized.