It depends. I like to travel light and don't carry a purse most days. When I do carry a purse, it'll be one of four that I have, depending on the situation.
For days when minimalist will do, I wear a festival belt. It's basically a stylish leather fanny pack with enough room for phone, small wallet, keys, lipstick, and a few tissues. It's great for any place where I want my hands free or there's a high risk of loss or theft.
Next up is a small leather crossbody bag I got from Next UK. It's big enough for phone, larger wallet, keys, small makeup kit, a couple of panty liners, tissues, and assorted bits.
A few months ago, I bought a larger theft-resistant crossbody bag expressly for travel. It sports slash-resistant fabric, cut-resistant straps, lockable zippers, and RFID protection. The bag is large enough for phone, keys, larger wallet, larger makeup kit, panty liners, tissues, mints, hand sanitizer, wipes, water bottle, and my small mirrorless camera with 1-2 lenses.
My largest bag is also my nicest. It's an Italian leather crossbody tote. I mostly use it when I'm wearing heels because it's big enough to carry a spare pair of flats. There's plenty of room for my phone, keys, large wallet, large makeup kit, panty liners, tissues, band-aids/moleskin, mints, hand sanitizer, wipes, water bottle, folding umbrella, and my small mirrorless camera with 1-2 lenses. The bag is so big that weight can be an issue when it's full.