in reference to the 1st generation Apple Watch

Hi

Yes, you can - potentially including when the paired iPhone is neither turned on nor nearby.

For Apple Watch (1st generation), the fullest range of features is available when Apple Watch and the paired iPhone are connected via Bluetooth and your iPhone has access to both cellular service and the Internet (via either cellular data or Wi-Fi).

You do not necessarily need to have your iPhone on or immediately around you (within arm's reach, for example) for the devices to establish and maintain their Bluetooth connection. Normal Bluetooth range is around 33 feet / 10 meters (this will vary in practice due to wireless interference).

Whenever your watch cannot connect to the iPhone via Bluetooth, it will attempt, as a fallback, to connect to a known Wi-Fi network - this being a compatible 802.11b/g/n 2.4GHz Wi-Fi network (some exceptions apply) that your iPhone has connected to before whilst your Apple Watch was connected to the iPhone via Bluetooth.

When instead connected to Wi-Fi, the watch also still offers a number of features, including receiving and sending messages via iMessage. If your carrier offers Wi-Fi Calling and it is supported on your plan and enabled on your iPhone, you can also use Apple Watch to make and receive phone calls and (subject to carrier support) to send and receive SMS / text messages.

If your Apple Watch is unable to connect either via Bluetooth or Wi-Fi, it again still offers some - although fewer - features.

Features that are available in each case are detailed here:

How to use your Apple Watch without your iPhone nearby - Apple Support

For the best performance from your iPhone and Apple Watch, including the most power-efficient communication between them, Apple recommends keeping both Bluetooth and Wi-Fi permanently enabled on your iPhone.

Keep in mind that Apple Watch (1st generation) cannot be used at all (even for telling the time) without being paired to a compatible model of iPhone. An iPhone 5s or newer model of iPhone is required, running the latest version of iOS 11.x. This is required in order to set up and use the Apple Watch, including for managing and syncing with the Apple Watch over time:

Set up your Apple Watch - Apple Support

More information and instructions:

• About Bluetooth and Wi-Fi on Apple Watch - Apple Support
• Answer phone calls
• Make phone calls
• Make a call with Wi-Fi Calling - Apple Support
• Make calls over Wi-Fi

The above answer is incorrect. All models of Apple watch can answer calls.

The Series 3 GPS+Cellular model does not require an iPhone nearby. All other models require an iPhone to be turned on and nearby to get a call that can be answered on the watch.

Read here for details:

WatchOS User Guide - Calls - Apple Support.

You cannot answer phone calls from an Apple Watch unless you have an Apple Watch Series 3.

Oh, well, you can answer phone calls that come in from a nearby iPhone when they're paired.

You're welcome! 🙂