Some American states have transitioned to universal voting-by-mail, where all registered voters receive a ballot in the mail. While this practice was growing in popularity prior to the 2020 general election, universal voting-by-mail was suddenly used in a larger number of states due to the COVID-19 pandemic. In this paper, we utilize a unique situation in which registered voters in some legislative districts in Los Angeles County were subjected to universal voting-by-mail in the March 2020 primary, while most of the rest of the Los Angeles County electorate was not. Using difference-in-difference and regression discontinuity designs, we estimate the causal effects of universal voting-by-mail on voter turnout and on who votes. Our results indicate that voter turnout increased by around 3\% for voters who do not automatically receive a ballot in the mail otherwise, and the increase is larger for registered partisan voters than those without a party affiliation.