According to a theory that is gaining increasing support, we should expect large asset managers (and, in particular, index fund managers) to become “climate stewards” and force companies to reduce their impact on climate change. According to this theory, by maximizing the value of their entire portfolio (portfolio primacy) rather than the value of the individual company (shareholder primacy), index fund managers are incentivized to reduce climate externalities and therefore to steer companies towards decarbonization. This Article offers the first systematic critique of this theory and identifies four crucial limits that undermine its practical impact: portfolio biases, mispricing of climate mitigation, fiduciary failures, and insulation from index funds’ influence. First, although large index funds are often referred to as “universal owners,” they only invest in subsets of the economy that are relatively less vulnerable to climate change. I show that index fund portfolios are overexposed to the oil & gas sector, to mega-capitalization companies, and to rich countries, and they are underexposed to small companies and to emerging or developing economies. Hence, index funds internalize global climate externalities in a very limited and imperfect way. Second, the stock market underestimates the social benefits of climate mitigation for two main reasons: stock prices do not accurately incorporate climate risk, and private investors discount the distant future at a much higher rate than the social discount rate. As a consequence, index funds should be expected to massively underinvest in climate mitigation. Third, even if index funds’ clients did benefit from climate stewardship, fund managers would have weak incentives to engage in it due to agency problems, collective action problems, and fiduciary conflicts inherent to their business model. Furthermore, to the extent that fund managers act on behalf of their clients, they fail to represent the interests of other constituencies who are even more vulnerable to climate change. Fourth, even if index fund managers undertook the role of climate stewards, most firms across the world would be partially or totally insulated from their influence, whether because they are privately held, are owned by state governments, or have a controlling or influential shareholder. The analysis of this Article reveals the serious limits of portfolio primacy and shows that this approach offers no adequate answer to the crucial threat of climate change. Policymakers, market participants, and concerned citizens should be warned that the promise of portfolio primacy to address climate risk is grossly overstated.