Data used to train machine learning (ML) models can be sensitive. Membership inference attacks (MIAs), attempting to determine whether a particular data record was used to train an ML model, risk violating membership privacy. ML model builders need a principled definition of a metric to quantify the membership privacy risk of (a) individual training data records, (b) computed independently of specific MIAs, (c) which assesses susceptibility to different MIAs, (d) can be used for different applications, and (e) efficiently. None of the prior membership privacy risk metrics simultaneously meet all these requirements. We present SHAPr, a membership privacy metric based on Shapley values which is a leave-one-out (LOO) technique, originally intended to measure the contribution of a training data record on model utility. We conjecture that contribution to model utility can act as a proxy for memorization, and hence represent membership privacy risk. Using ten benchmark datasets, we show that SHAPr is indeed effective in estimating susceptibility of training data records to MIAs. We also show that, unlike prior work, SHAPr is significantly better in estimating susceptibility to newer, and more effective MIA. We apply SHAPr to evaluate the efficacy of several defenses against MIAs: using regularization and removing high risk training data records. Moreover, SHAPr is versatile: it can be used for estimating vulnerability of different subgroups to MIAs, and inherits applications of Shapley values (e.g., data valuation). We show that SHAPr has an acceptable computational cost (compared to naive LOO), varying from a few minutes for the smallest dataset to ~92 minutes for the largest dataset.