Histone methyltransferase that specifically trimethylates 'Lys-9' of histone H3 using monomethylated H3 'Lys- 9' as substrate. Also weakly methylates histone H1 (in vitro). H3 'Lys-9' trimethylation represents a specific tag for epigenetic transcriptional repression by recruiting HP1 (CBX1; CBX3 and/or CBX5) proteins to methylated histones. Mainly functions in heterochromatin regions; thereby playing a central role in the establishment of constitutive heterochromatin at pericentric and telomere regions. H3 'Lys-9' trimethylation is also required to direct DNA methylation at pericentric repeats. SUV39H1 is targeted to histone H3 via its interaction with RB1 and is involved in many processes; such as repression of MYOD1-stimulated differentiation; regulation of the control switch for exiting the cell cycle and entering differentiation; repression by the PML-RARA fusion protein; BMP-induced repression; repression of switch recombination to IgA and regulation of telomere length. Component of the eNoSC (energy-dependent nucleolar silencing) complex; a complex that mediates silencing of rDNA in response to intracellular energy status and acts by recruiting histone- modifying enzymes. The eNoSC complex is able to sense the energy status of cell: upon glucose starvation; elevation of NAD(+)/NADP(+) ratio activates SIRT1; leading to histone H3 deacetylation followed by dimethylation of H3 at 'Lys-9' (H3K9me2) by SUV39H1 and the formation of silent chromatin in the rDNA locus. Recruited by the large PER complex to the E-box elements of the circadian target genes such as PER2 itself or PER1; contributes to the conversion of local chromatin to a heterochromatin-like repressive state through H3 'Lys-9' trimethylation. {ECO:0000269|PubMed:14765126; ECO:0000269|PubMed:16449642; ECO:0000269|PubMed:16818776; ECO:0000269|PubMed:16858404; ECO:0000269|PubMed:18004385; ECO:0000269|PubMed:18485871}.