On 13 March 1780, Alexander and HMS Courageaux captured the 40-gun French privateer Monsieur after a long chase and some exchange of fire.

[3] In 1794, whilst returning to England in the company of HMS Canada after escorting a convoy to Spain, Alexander, under the command of Rear-Admiral Richard Rodney Bligh, fell in with a French squadron of five 74-gun ships, and three frigates, led by Joseph-Marie Nielly.

In 1798 Nelson was detached into the Mediterranean by Earl St. Vincent with HMS Orion, Alexander, Emerald, Terpsichore, and Bonne Citoyenne.

They sailed from Gibraltar on 9 May and on 12 May were struck by a violent gale in the Gulf of Lion that carried away Vanguard's topmasts and foremast.

[citation needed] Northumberland, Alexander, Penelope, Bonne Citoyenne, and the brig Vincejo shared in the proceeds of the French polacca Vengeance, captured entering Valletta, Malta on 6 August.

[5] Alexander served in the navy's Egyptian campaign between 8 March 1801 and 2 September, which qualified her officers and crew for the clasp "Egypt" to the Naval General Service Medal that the Admiralty issued in 1847 to all surviving claimants.