Let f (x; y) (f (x)) be a symmetric real function (real function) and G = (V; E) be a graph. Denote by di the degree of a vertex i in G. The graphical function-index T If (G) (Hf (G)) of G with edge-weight (vertex-weight) function f (x; y) (f (x)) is defined as T If (G) = Puv2E f (du; dv) (Hf (G) = Pu2V f (du)). We can also get a weighted adjacency matrix from the edge-weighted graph, i.e., Af (G) = (aifj) where aifj = f (di; dj) if vertices i and j are adjacent in G, and 0 otherwise. This matrix is simply referred to as the f -weighted adjacency matrix. One can see that the concepts of graphical function-indices and f -weighted adjacency matrix can cover all the degree-based graphical indices and degree-based adjacency matrices of graphs, such as the Zagreb indices, Randic´ index, ABC-index, etc., and the Randic´ matrix, ABC-matrix, GA-matrix, etc. So, for the graphical function-indices T If (G) and Hf (G) and the f -weighted adjacency matrix Af (G) of a graph G, one can think about finding unified ways to study the extremal problems and spectral problems. This survey is intended to sum up the results done so far on these problems.