In this article, we consider p-adic modeling of the standard genetic code and the vertebrate mitochondrial one. To this end, we use 5-adic and 2-adic distance as a mathematical tool to describe closeness (nearness, similarity) between codons as elements of a bioinformation space. Codons which are simultaneously at the smallest 5-adic and 2-adic distance code the same (or similar) amino acid or stop signal. The set of codons is presented as an ultrametric tree as well as a fractal and p-adic network. It is shown that genetic code can be treated as sequential translation between genetic languages. This p-adic approach gives possibility to be applied to sequences of nucleotides of an arbitrary finite length. We present an overview of published and some new results on various p-adic properties of the genetic code.