Learning with limited labelled data, such as few-shot learning, meta-learning or transfer learning, aims to effectively train a model using only small amount of labelled samples. However, these approaches were observed to be excessively sensitive to the effects of uncontrolled randomness caused by non-determinism in the training process. The randomness negatively affects the stability of the models, leading to large variance in results across training runs. When such instability is disregarded, it can unintentionally, but unfortunately also intentionally, create an imaginary perception of research progress. Recently, this area started to attract a research attention and the number of relevant studies is continuously growing. In this survey, we provide a comprehensive overview of 134 papers addressing the effects of randomness on the stability of learning with limited labelled data. We distinguish between four main tasks addressed in the papers (investigate/evaluate; determine; mitigate; benchmark/compare/report randomness effects), providing findings for each one. Furthermore, we identify and discuss seven challenges and open problems together with possible directions to facilitate further research. The ultimate goal of this survey is to emphasise the importance of this growing research area, which so far has not received appropriate level of attention.