Classes:
I: Double-stranded DNA (Herpesviruses; Poxviruses, etc.

• Usually must enter the host nucleus before it is able to replicate
• Require host cell polymerases to replicate the viral genome--> highly dependent on the cell cycle
• The virus may induce the cell to forcefully undergo cell division--> Causes cancer

There is only one well studied example in which a class 1 virus is not replicating within the nucleus, and that is the Poxvirus family, a highly pathogenic virus that infects vertebrates and includes the smallpox virus. They replicate in the cytoplasm and make their own enzymes for nucleic acid replication.

II: Single-stranded (+)sense DNA (Parvoviruses)Replication occurs in the nucleus, involving the formation of a (-)sense strand, which serves as a template for (+)strand RNA and DNA synthesis. Viruses that fall under this category include ones that are not as well studied, but still pertain highly to vertebrates.

III: Double-stranded RNA (Reoviruses; Birnaviruses)These viruses have segmented genomes. Each genome segment is transcribed separately to produce monocistronic mRNAs. As with most RNA viruses, this class replicates in the cytoplasm, not having to use the host replication polymerases as much as DNA viruses.

IV: Single-stranded (+)sense RNA (Picornaviruses etc.)a) Naked RNA is infectious, no virion particle associated polymerase. Translation results in the formation of a polyprotein product, which is subsequently cleaved to form the mature proteins.b) Complex Transcription e.g. Togaviruses. Two or more rounds of translation are necessary to produce the genomic RNA.

V: Single-stranded (-)sense RNA (Orthomyxoviruses etc.)Must have a virion particle RNA directed RNA polymerase.a) Segmented e.g. Orthomyxoviruses. First step in replication is transcription of the (-)sense RNA genome by the virion RNA-dependent RNA polymerase to produce monocistronic mRNAs, which also serve as the template for genome replication.b) Non-segmented e.g. Rhabdoviruses. Replication occurs as above and monocistronic mRNAs are produced.

VI: Single-stranded (+)sense RNA with DNA intermediate in life-cycle (Retroviruses)Genome is (+) sense but unique among viruses in that it is diploid, and does not serve as mRNA, but as a template for reverse transcription to convert the positive sense RNA into DNA. Instead of using the RNA for templates of proteins, they use DNA to create the templates, which is spliced into the host genome using integrase. Replication can then commence with the help of the host cell's polymerases.

VII: Double-stranded DNA with RNA intermediate (Hepadnaviruses)This group of viruses also relies on reverse transcription, but unlike the Retroviruses, this occurs inside the virus particle on maturation. On infection of a new cell, the first event to occur is repair of the gapped genome subsequently filled in to form a covalently closed circle , followed by transcription. The pregenome RNA serves as template for the viral reverse transcriptase and for production of the DNA genome.

Read the whole paper submitted by David Baltimore here.

For a brief overview of this topic, here are pictures that summarizes what is mentioned above: