  In this work we study symmetric random matrices with variance profile
satisfying certain conditions. We establish the convergence of the operator
norm of these matrices to the largest element of the support of the limiting
empirical spectral distribution. We prove that it is sufficient for the entries
of the matrix to have finite only the $4$-th moment or the $4+\epsilon$ moment
in order for the convergence to hold in probability or almost surely
respectively. Our approach determines the behaviour of the operator norm for
random symmetric or non-symmetric matrices whose variance profile is given by a
step or a continuous function, random band matrices whose bandwidth is
proportional to their dimension, random Gram matrices, triangular matrices and
more.
