Compositions that are used in nucleic acid amplification in vitro are disclosed, which include a target specific universal (TSU) promoter primer or promoter provider oligonucleotide that includes a target specific (TS) sequence that hybridizes specifically to a target sequence that is amplified and a universal (U) sequence that is introduced into the sequence that is amplified, by using a primer for the universal sequence. Methods of nucleic acid amplification in vitro are disclosed that use one or more TSU oligonucleotides to attached a U sequence to a target nucleic acid in a target capture step and then use a primer for a U sequence in subsequent amplification steps performed in substantially isothermal conditions to make amplification products that contain a U sequence that indicates the presence of the target nucleic acid in a sample.