While older versions of the default Android camera app include the photo sphere mode (letting you take 360 images in your regular camera app), this feature was unfortunately discontinued in recent years. This makes taking photos with your mobile harder than it used to be.
On Android, Go Street View Photo Sphere works well, although it can struggle with very close objects. While it’s designed with creating images for Street View in mind, it does a good job of creating 360 panoramic photos that can be saved directly to your device.
On Apple, apps such as 360 Photo Cam are available.
Note that capturing using mobile phone will usually have worse results – particularly with close-up features – than using a true 360-degree camera. This is because image-stitching systems often have difficulty dealing with close-up objects when building a panoramic image from multiple still images.
You can also capture footage using dedicated 360 cameras and phone attachments.