Question: I'm wondering what the difference is, and at what stage these different terms are used within the Agile/Scrum methodology?

`use case`, `user requirements` and `user stories`

Am I right in thinking they start off in the requirement elicitation stage as `use cases` then become the `user requirements`, which are pretty much the same as the `user stories` used in the `products backlog`?

Trying to write a university report and haven't been able to find anything defining all of these terms in the same place, which has made it difficult to understand the differences.

Are some of these terms even required within an Agile approach?

Answer: 
**Use Case** - Formal specification of interaction between actor(s) and a system that realize one single functional requirement of this system (part of UML).

**User Story** - Informal description of a function of a system (Agile term). 

**User Requirement** - Formal description of what user expects from the system usually gathered/formulated on very early stage of the development process by an architect in close cooperation with the client. There are two main types of requirements:

1. **Functional** - requirements that are translated to use cases or user stories and then implemented in business logic of the application.
2. **Non-functional** - requirements that (in most cases) should be ensured by the architecture of the system. It's very important that all non-functional requirements are formulated in measurable way, so they can be tested.

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It makes less sense to compare User Requirements to User Stories or Use Cases, because requirements serve very different purpose. Use Cases and User Stories describe details of a function in the system, while User Requirements state functions and non-functional properties of the system (unambiguously but without any detail).
In simple words: `user stories` are not `use cases`, and `user requirements` are more fine-grained document of requirements that are usually approved (or pre-approved) by client.

The user stories **don’t provide** the details the team needs to do their work. The Scrum process enables this detail to emerge organically, mainly by removing the need to write use cases.

Also there’s consensus that use cases and user stories are not interchangeable:

* Alistair Cockburn: A user story is to a use case as a gazelle is to a gazebo
* ExtremeProgramming.org: User stories serve the same purpose as use cases but are not the same.
* Mike Cohn: User stories aren’t use cases

You may also thing about **user stories** as - a short description of something that your customer will do when they come to your website or use your application/software, focused on the value or result they get from doing this thing. They are written from the point of view of a person using your website or application, and written in the language that your customers would use. 

Meanwhile, **use cases** are more a description of a set of interactions between a system and and one or more actors (where ‘actor’ can be people, or other systems: for example, both online shoppers and PayPal can be actors). They are usually created as documents

**References**:

* [User Story And Use Case Comparison](http://c2.com/cgi/wiki?UserStoryAndUseCaseComparison)
* [Linkedin discussion: What is a difference between Use Cases and User Stories in agile development?](https://www.linkedin.com/groups/What-is-difference-between-Use-37631.S.233728596)