Evolution is marked by well-defined events involving profound innovations that are known as ‘major evolutionary transitions'. They involve the integration of autonomous elements into a new, higher-level organization whereby the former isolated units interact in novel ways, losing their original autonomy. All major transitions, which include the origin of life, cells, multicellular systems, societies or language (among other examples), took place millions of years ago. Are these transitions unique, rare events? Have they instead universal traits that make them almost inevitable when the right pieces are in place? Are there general laws of evolutionary innovation? In order to approach this problem under a novel perspective, we argue that a parallel class of evolutionary transitions can be explored involving the use of artificial evolutionary experiments where alternative paths to innovation can be explored. These ‘synthetic’ transitions include, for example, the artificial evolution of multicellular systems or the emergence of language in evolved communicating robots. These alternative scenarios could help us to understand the underlying laws that predate the rise of major innovations and the possibility for general laws of evolved complexity. Several key examples and theoretical approaches are summarized and future challenges are outlined. This article is part of the themed issue ‘The major synthetic evolutionary transitions’.