Nonlocal transformations of a number of the quasilinear parabolic equations describing spherically symmetrical heat conduction and diffusion processes are considered. One of them transforms the equation $r^{n-1}\theta_r=(r^{n-1}|\theta_r|^l\theta_r)_r$ into equation of the same type but with another value of the exponent $n$. Other transformation converts the equation $r^{n-1}\theta_t=(r^{n-1}\theta^{-2}\theta_r)_r$ into equation whose coefficients do not depend on space variable. The third nonlocal transformation holds invariant the equation $r\theta_r=(r\theta^{-1}\theta_r)_r$. Some exact solutions of the mentioned equations are analysed incidentally. Bibliography: 15 titles.