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Summary: Background: Obstructive Sleep Apnoea (OSA) often co-occurs with cardiometabolic and pulmonary diseases. This study is to apply genetic analysis methods to explain the associations between OSA and related phenotypes. Methods: In the Hispanic Community Healthy Study/Study of Latinos, we estimated genetic correlations ρg between the respiratory event index (REI) and 54 anthropometric, glycemic, cardiometabolic, and pulmonary phenotypes. We used summary statistics from published genome-wide association studies to construct Polygenic Risk Scores (PRSs) representing the genetic basis of each correlated phenotype (ρg>0.2 and p-value<0.05), and of OSA. We studied the association of the PRSs of the correlated phenotypes with both REI and OSA (REI≥5), and the association of OSA PRS with the correlated phenotypes. Causal relationships were tested using Mendelian Randomization (MR) analysis. Findings: The dataset included 11,155 participants, 31.03% with OSA. 22 phenotypes were genetically correlated with REI. 10 PRSs covering obesity and fat distribution (BMI, WHR, WHRadjBMI), blood pressure (DBP, PP, MAP), glycaemic control (fasting insulin, HbA1c, HOMA-B) and insomnia were associated with REI and/or OSA. OSA PRS was associated with BMI, WHR, DBP and glycaemic traits (fasting insulin, HbA1c, HOMA-B and HOMA-IR). MR analysis identified robust causal effects of BMI and WHR on OSA, and probable causal effects of DBP, PP, and HbA1c on OSA/REI. Interpretation: There are shared genetic underpinnings of anthropometric, blood pressure, and glycaemic phenotypes with OSA, with evidence for causal relationships between some phenotypes. Funding: Described in Acknowledgments.