This article features some reflections by two senior statisticians/data scientists on the challenges arising from the analysis of increasingly complex data using robustness. Robustness has a long history in statistics and recognizes that all statistical models are simplifications of the reality of the data and it is important to have procedures which robustly ensure that our conclusions remain valid across deviations from the model) assumptions. We present our thoughts on how to approach the analysis of the evermore complex data currently collected. The aim of much data analysis in Biology and Medicine is to detect a weak signal in noisy data. Robust procedures are ideally suited to find and identify anomalous structures in the data which deviate from the bulk of the data In studies of cancer and other diseases, the interest may well be in this anomalous data. We also give some recent examples of research problems we worked on recently where robustness played a very useful role. We include some thoughts on the challenges that robust analysis which)will be facing in the next few years cognizant of our very limited ability to successfully predict the future.