The evolutionary trajectories of biological sequences are propelled by mutation and whittled away by selection to maintain function. Present day sequences can therefore be regarded as the outcomes of millions of evolutionary experiments that are shadows of underlying constraints in the genotype-phenotype map. When examining a single position in a sequence, this signal of 'conservation' is staple piece information for molecular biology and biotechnology, but it is hard to attribute conservation at a single site to a cause. In this talk I will introduce computational methods that, when combined with recent growth in sequence databases, make it possible to explain evolutionary conservation in terms of evolutionary couplings between residues. We have applied these tools to predict (i) accurate 3D structures of protein, RNA and complexes, (ii) conformational plasticity of 'disordered' proteins and (iii) the quantitative effects of mutations on organism fitness. These computational approaches address the challenge of inferring causality from correlations in biological information such as sequences but can be applied more widely to other biological information such as gene expression or dynamics, cellular phenotypes, drug response. I will introduce challenges and opportunities for extending these methods to diverse proteins and RNAs and developing them for biomedical and engineering applications.