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Let $F_2$ denote the finite field of two elements. $An [n,k]$  binary linear block code $C$ is a $k$-dimensional subspace of $F_2^n$. For $u,v∈F_2^n$, let $d(u,v)$ denote the number of coordinate places where $u$ and $v$ differ, which is called the Hamming distance between $u$ and $v$. The covering radius of a block code $C$ of length $n$ is the smallest integer $R=R(C)$ such that all vectors in the containing space are within Hamming distance $R$ of some codeword of $C$. Equivalently, the covering radius of $C$  is the smallest integer $R$ such that every $q$-ary $(n −k)$ tuple can be written as a linear combination of at most $R$ columns of the parity-check matrix of $C$. There has been an intense interest in covering radius since a paper of Delsarte in 1973. It has applications to problems of data compression, testing, and write-once memories etc. Computing the covering radius of a given code is a hard task. Cyclic codes are one of the most commonly used class of linear block codes, where the circular shifts of each codeword gives another word that belongs to the code. They have a rich algebraic structure which are useful for efficient error detection and correction. In this talk, we will mention from some results regarding the  covering radii of cyclic codes over binary finite fields.