A method, implemented by a computer-implemented authentication system, for authenticating a user attempting to access a target component of a computer system from a first user system, the first user system performing a login process and being connectable to a first communications network, the method comprising: - receiving user authentication information entered by the user via a first user-interface of the first user system during the login process; - obtaining via the first communications network a session identifier identifying a login session, and at least one item of session-specific contextual information collected by the login process, the session-specific contextual information conveying user-verifiable information about a context of the login operation, the session-specific contextual information being chosen from information indicative of the first user system or of a property of the first communications network; - verifying the user authentication information and, responsive to successful verification of the user authentication information, - generating, during the login session, a real-time session-specific one-time passcode associated with the session identifier and unique for the login session; - sending a message from the computer-implemented authentication system via a second communications network to a second user system assigned to the user, the message comprising: a) the at least one item of said obtained session-specific contextual information or derived information of said obtained session-specific contextual information, allowing the second user system to present at least the at least one item of session-specific contextual information to the user via a second user interface so as to allow the user to verify the at least one item of session-specific contextual information, and b) the generated real-time session-specific one-time passcode; - receiving, via the first user system and the first communications network, a passcode entered by the user into the first user system; - verifying the entered passcode against the real-time session-specific one-time passcode and/or the session identifier; and, responsive to a successful verification of the entered passcode, granting to the first user system access to the target component. - receiving user authentication information entered by the user via a first user-interface of the first user system during the login process; - obtaining via the first communications network a session identifier identifying a login session, and at least one item of session-specific contextual information collected by the login process, the session-specific contextual information conveying user-verifiable information about a context of the login operation, the session-specific contextual information being chosen from information indicative of the first user system or of a property of the first communications network; - verifying the user authentication information and, responsive to successful verification of the user authentication information, - generating, during the login session, a real-time session-specific one-time passcode associated with the session identifier and unique for the login session; - sending a message from the computer-implemented authentication system via a second communications network to a second user system assigned to the user, the message comprising: a) the at least one item of said obtained session-specific contextual information or derived information of said obtained session-specific contextual information, allowing the second user system to present at least the at least one item of session-specific contextual information to the user via a second user interface so as to allow the user to verify the at least one item of session-specific contextual information, and b) the generated real-time session-specific one-time passcode; a) the at least one item of said obtained session-specific contextual information or derived information of said obtained session-specific contextual information, allowing the second user system to present at least the at least one item of session-specific contextual information to the user via a second user interface so as to allow the user to verify the at least one item of session-specific contextual information, and b) the generated real-time session-specific one-time passcode; - receiving, via the first user system and the first communications network, a passcode entered by the user into the first user system; - verifying the entered passcode against the real-time session-specific one-time passcode and/or the session identifier; and, responsive to a successful verification of the entered passcode, granting to the first user system access to the target component.