'Could' is one of the most commonly known modals verbs. It helps us talk about abilities, ask for permission, request something, offer things, etc. We use it to refer to the past. In other words, 'could' is the past tense of modal verb 'can.' Have a look:

'May' is another modal verb manly used to talk about possibilities, give and ask for permissions, etc. Consider the following examples:

We use 'could' and 'may' are used to give and ask for permission.

Both 'may' and 'could' are used to give permission. 'May' is much politer and more formal than 'could.' Compare:

'May' and 'could' are used in the present time to ask for permission. 'May' is politer than 'could' and therefore it is mainly used when talking to authorities. For example:

We use 'may' and 'could' to refer to something we are not certain about their chance of occurrence.
'Could' is used to refer to past, present, and future events that had or still have a chance of happening.

'May' is also used to show a chance of occurrence in the present and the future.

Offers are statements that help us show that we are ready and willing to do something for someone. We use 'could' and 'may' to make such offers.
'Could' is not past tense, when we are offering something. Offers that are made with 'could' are less definite than other forms of offers. This means that the offer is made but we are uncertain if the listener will accept or not.
When making offers with 'may,' we use a first-person singular or plural pronoun (I and we).
Note that 'may' is more formal than 'could.'
For instance:

As modal verbs, 'could' and 'may' are subjects of change when creating negative sentences. To make them negative, we add 'not' to them as shown below:

Here are some examples:

Modal verbs are inverted when we are creating a question. Watch:

We only have one modal verb in a sentence and we cannot add more than one modal verb to a sentence. Have a look:

We use the modal verb 'could' to give advice and express our opinion on how something could be better. We often use 'could' to show our disapproval of something. Since 'could' is past tense, it means that situation we are talking about has already happened and there is no way of changing it. For instance:

We use 'may' to talk about our wishes, prayers, or hopes. In this function, 'may' is often placed at the beginning of the sentence. For instance:

We use 'could' to talk about past abilities. This means that the ability, talent, or skill existed before but it no longer exists or is not used anymore. Take a look at the following examples:

'May' is more formal than 'could.' In formal contexts, it is advised to use 'may' to show gratitude and respect. Have a look:

'Could' and 'may' are modal verbs that can be used as conditional verbs.

'May' and 'could' are both used in conditionals type 1. In this type, we show a condition and its results. These conditions are real situations with a high chance of occurrence.

Conditional type 2 talks about hypothetical situations. These situations are imaginary and unlikely to happen in the present or future. The chance of these situations becoming real is very low. We use the modal verb 'could' to talk about these situations.

Conditional type 3 is the only conditional that talks about the past. This type talks about hypothetical situations in the past. In other words, we are describing a different past or a different outcome from past events.