Context: Software start-ups are emerging as suppliers of innovation and software-intensive products. However, traditional software engineering practices are not evaluated in the context, nor adopted to goals and challenges of start-ups. As a result, there is insufficient support for software engineering in the start-up context. Objective: We aim to collect data related to engineering goals, challenges, and practices in start-up companies to ascertain trends and patterns characterizing engineering work in start-ups. Such data allows researchers to understand better how goals and challenges are related to practices. This understanding can then inform future studies aimed at designing solutions addressing those goals and challenges. Besides, these trends and patterns can be useful for practitioners to make more informed decisions in their engineering practice. Method: We use a case survey method to gather first-hand, in-depth experiences from a large sample of software start-ups. We use open coding and cross-case analysis to describe and identify patterns, and corroborate the findings with statistical analysis. Results: We analyze 84 start-up cases and identify 16 goals, 9 challenges, and 16 engineering practices that are common among start-ups. We have mapped these goals, challenges, and practices to start-up life-cycle stages (inception, stabilization, growth, and maturity). Thus, creating the progression model guiding software engineering efforts in start-ups. Conclusions: We conclude that start-ups to a large extent face the same challenges and use the same practices as established companies. However, the primary software engineering challenge in start-ups is to evolve multiple process areas at once, with a little margin for serious errors.