Basemap now lets you assign specific Project Tasks to your project's team members, both when creating the Task initially and later as responsibility for a Task changes from person to person.
To assign a Task to a project team member when you're creating it, first click the "Show More" button next to the Task quick entry field. This expands the interface to let you provide a description and due date, as well as choose the assignee for your new Task.
Click the arrow next to the "Assign..." field to view the list of Project users and select the new Task's assignee.
When you're viewing the list of Tasks in your Project, hovering over an individual Task gives you a couple of important options. The pencil icons that appear let you edit each of the fields for your task. Clicking the pencil icon to the right of the current assignee for a Task lets you re-assign that Task to another user in the project.
To the left, you can click the "i" Info icon to view detailed information and history for the Task. This includes when it was created, which user created it, when it was last modified, who modified it, and the history of changes to its assignment. Note that "Last Modified" shows when a change was last made to a Task's title, description, or due date — that is, a change to the purpose or priority of the task. The Last Modified date doesn't change if the only modification to a Task is its assignment.