Let X be a Banach space, and L(X) be the space of bounded linear operators from X into X. B1(X) denotes the closed unit ball of X, and S1(X )i s unit sphere of X. An element T ∈ S1(L(X)) is called extreme operator if there is no A ∈ L(X) such thatT ± A �≤ 1. The set of extreme points of S1(X) will be denoted by ext(S1(X)) .T ∈ S1(L(X)) is called nice if T ∗ (ext(S1(L(X ∗ ))) ⊆ ext(S1(L(X ∗ ))). The object of this paper is to give simpler proofs of old results and present new results on extreme operators of S1(L(� 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 onp −spaces are characterized. I. Introduction. Let X be a Banach space. The closed unit ball of X will be denoted by B1(X) , and the unit sphere by S1(X). An element x ∈ S1(X) is called an extreme point of B1(X) if whenever x = 1 (y + z), with y and z in S1(X), then x = y = z. The space of bounded linear operators on X will be denoted by L(X), and the compact ones by K(X). Extreme elements of S1(L(X)) are called extreme operators. An operator T ∈ L(X )i s called nice if the set of extreme points of B1(X ∗ ) is an invariant set for T ∗ . L p (I), 1 ≤ p< ∞, denotes the Banach space of p-Bochner integrable functions (equiv- alence classes) defined on the unit interval, with the usual classical norm. Similarly, � p denotes the Banach space of p-summable sequences, with the usual classical norm. The space of continuous functions on a compact set Ω, with the uniform norm is denoted by C(Ω).