Eukaryotic genes contain non-coding introns, removal of which during gene expression is a pre-requisite for gene function. Removal of introns and ligation of coding exons ––a process called splicing––is catalysed by a dynamic macromolecular machine called the spliceosome. The spliceosome assembles anew on precursor mRNA (pre-mRNA) from its component small nuclear ribonucleoproteins (snRNPs) and protein factors. A series of conformational and compositional changes produces the catalytically active spliceosome, in which the snRNA components of the snRNPs mediate intron recognition and catalysis of splicing. Despite the importance of splicing to fundamental eukaryotic molecular biology, and the contribution alternative splicing makes to organismal diversity and disease aetiology, a structural view of the spliceosome during catalysis has been lacking. Therefore, it is unclear how the spliceosome’s active site is formed and stabilised, how protein factors contribute to the chemistry, fidelity, and dynamics of splicing, and exceptionally how the 3′ splice site—i.e. the end of an intron—is recognised. This thesis first describes two atomic structures of the spliceosome from budding yeast, stalled immediately after each chemical reaction of splicing and resolved by electron cryomicroscopy. The structure of the spliceosome immediately after branching shows how the 5′ splice site is ligated to the branch point adenosine upstream of the 3′ splice site. It reveals the three-dimensional conformation of the RNA-based active site, its stabilisation by tertiary interactions and an extensive protein scaffold, and how the branching factors Cwc25, Yju2, and Isy1 promote docking of the branch point adenosine into the active site. The position of the Prp16 ATPase gave the first structural insight into how the DExD/H-box helicase proteins might remodel ribonucleoprotein complexes. The structure of the spliceosome immediately after exon ligation shows the connected 5′ and 3′ exons and how the active site is remodelled to allow two different reactions in a single active site. It shows large-scale conformational changes in the spliceosome associated with exon ligation and how these might be stabilised by the exon-ligation factors Prp18, Slu7, and Prp17. Most importantly it shows how the AG dinucleotide defining the 3′ splice site is recognised by non-Watson-Crick base pairs to the branch point adenosine and 5′ splice site guanosine. This explains the importance of the most conserved intron nucleotides and justifies a branching mechanism for splicing. Finally, it is unclear how the 5′ splice site, which is initially recognised by the U1 snRNP, is transferred into the spliceosome’s active site. The electron cryomicroscopy structure of a fully assembled pre-catalytic human spliceosome is described, stalled immediately before 5′ splice site transfer from U1 to U6 snRNA. The structure shows in high-resolution the human U4/U6.U5 tri-snRNP and how U6 snRNA is primed by RBM42 and SNRNP-27K for receiving the 5′ splice site. It shows how the DEAD-box ATPase Prp28 disrupts the 5′ splice site/U1 snRNA duplex and how this might be coupled to Brr2-mediated spliceosome activation.