The Gentleman's Magazine. London: W. Pickering, etc., 1731-1856.
Perkins/Bostock Microfilm: S5964s
Also available for online browsing for 1731-1750.
This magazine is unmatched in this library for the extent and duration of its reporting of births, marriages, deaths, and civil, military, and ecclesiastical appointments at home and abroad. The coverage was from 1731 to 1869 when the last of those features was discontinued. The indexing is excellent. Each volume or annual set has an index. General indices were compiled for 1731-1786 and 1787-1818 (shelved after 1786 and 1818). However, persons were often listed by last name only. A valuable supplement is the Index to the Biographical and Obituary Notices in the Gentlemen's Magazine from 1731 to 1780 (Perkins Reference AI3 .F377 1891) that provides given names as well as surnames (Vols. available here for A-MI only). Given names, initials, and titles were gradually incorporated into the indices of individual volumes, and those of later years are more informative. One caution: the larger obituaries and biographies were cited in the indices of essays and were not always included in the indices of names, so both should be consulted.

The Monthly Chronicle. London: A. Ward, etc., 1728-1732. 5 vols.
Perkins Special Collections: E M789C
Library has: Vols. 1-5, 1728-1732.
Included births, marriages, deaths, and civil, military, and ecclesiastical appointments in its "Chronicle" that was partially indexed annually, but the indices are to be found within the volumes rather than at their beginning or end. Continued as The London Magazine: Or, Gentleman's Monthly Intelligencer.

The London Magazine Or, Gentleman's Monthly Intelligencer. London: Printed by C. Ackers for Wilford etc., 1732-1785. 56 vols.
Special Collections Library, Library Service Center, Rare Books (Reading Room only)
Library has: 1732-1753, 1755-1785
Available in full-text at Eighteenth Century Collections Online.
Similar to the Gentleman's Magazine, but the entries are less numerous. Name indices in volumes by surname only. Supercedes The Monthly Chronicle.

The London Magazine. London: Printed for Baldwin, Cradock, and Joy, etc., 1820-1829. 23 vols.
Special Collections Library, Library Service Center, Rare Books (Reading Room only)
Library has: Vols. 1-10, 1820-1824
Similar to the Gentleman's Magazine and earlier London Magazine, but lacking indices.

The London Magazine and Monthly Critical and Dramatic Review. London: Gold and Northhouse, 1820-1829.
Special Collections Library, Library Service Center, Rare Books (Reading Room only)
Library has: Vols. 1-3, 1820- June 1821.
Includes births, marriages, and deaths. No indices. Also known as Gold's London Magazine and Theatrical Inquisitor.

European Magazine and London Review: Containing the Literature, History, Politics, Arts, Manners & Amusements of the Age. London: J. Sewell, 1782-1826.
Special Collections Library E-15A E89M
Library has: 1782-1791, 1793-1797, 1799-1816.
Regularly reported births, marriages, deaths, promotions, and bankrupts. Annual indices.

The Monthly Magazine, and British Register... London: Printed for R. Phillips, 1796-1823, 1824.
Special Collections Library E-6B M789
Library has: 1796-1823, 1824/25.
Monthly sections for events, marriages and deaths arranged by counties and London. Annual index of obituaries very limited. Annual index of ecclesiastical promotions.

The Town and Country Magazine, Or Universal Repository of Knowledge, Instruction, and Entertainment. London: Printed for A. Hamilton, Jr., 1769-1796. 28 vols.
Special Collections Library, Library Service Center, Rare Books (Reading Room only)
Library has: 1769-1772, 1774-1785.
Similar to The Gentleman's Magazine but less extensive. Coverage includes births, marriages, deaths, and bankruptcies. The volumes have name indices by surnames.