The covering radius of binary codes is studied. Bounds on K(n,R), the minimum cardinality of any binary code of length n and covering radius R, are found. Modifications of the van Wee lower bounds are proved for K(n,R), the minimal number of codewords in any binary code of length n and covering radius R. The first of the two van Wee bounds is based on studying the Hamming spheres of radius 1 centered at the points which have distance R to the code C. The points covered by more than one codeword are divided into several classes and better estimates for some of these classes are obtained. Using a suitable averaging process, the lower bound for K(n,R) when R>or=2 is improved. The second van Wee bound studies spheres of radius 2 centered at the points which have distance R-1 or R to the code C. These points are divided essentially into two classes: the points that are covered by only one codeword of C, and the points that are covered by more than one codeword. >