When do past participles agree in French?

When a verb in French uses a form of 'être' as an auxiliary alongside its past participle, the past participle agrees with the subject of the verb in gender and number. There is no agreement if the auxiliary is a form of 'avoir'. The only case where a verb in 'avoir' has an agreement is when there is a preceding direct object. This is when the direct object of a verb comes before the verb, rather than after it. In these cases, the past participle agrees with the preceding direct object in gender and number. For example, in the phrase 'j'ai vu la femme' the direct object 'la femme' comes after the verb - so there is no agreement. But in the phrase 'La femme que j'ai vue' the object 'la femme' precedes the verb, therefore the past participle 'vu' has an 'e' on the end in agreement with the feminine object.