The well-being of all living organisms relies on the accurate duplication of their genomes. This is usually achieved by highly elaborate replicase complexes which ensure that this task is accomplished timely and efficiently. However, cells often must resort to the help of various additional "specialized" DNA polymerases that gain access to genomic DNA when replication fork progression is hindered. One such specialized polymerase family consists of the so-called "translesion synthesis" (TLS) polymerases; enzymes that have evolved to replicate damaged DNA. To fulfill their main cellular mission, TLS polymerases often must sacrifice precision when selecting nucleotide substrates. Low base-substitution fidelity is a well-documented inherent property of these enzymes. However, incorrect nucleotide substrates are not only those which do not comply with Watson-Crick base complementarity, but also those whose sugar moiety is incorrect. Does relaxed base-selectivity automatically mean that the TLS polymerases are unable to efficiently discriminate between ribonucleoside triphosphates and deoxyribonucleoside triphosphates that differ by only a single atom? Which strategies do TLS polymerases employ to select suitable nucleotide substrates? In this review, we will collate and summarize data accumulated over the past decade from biochemical and structural studies, which aim to answer these questions.