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If I want to connect a LiFePO4 battery (as the power supply) directly to an esp32 - would this be the correct way: the plus of the battery to the 3.3 volt pin and the minus to a ground pin?

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No. Absolutely not.

A LiFePO4 battery is 4.2V when fully charged. The absolute maximum voltage of an ESP32 is 3.6V.

You will kill the ESP32.

Instead you need to have a voltage regulator with a very low dropout. Ideally a switching ("buck") regulator with a zero-dropout "pass" mode to give the maximum life from your battery.

Even better is a "buck/boost" battery regulator that will boost the voltage up to a stable 3.3V once the battery voltage drops too low.

If all this is beyond you then you're better off using a powerbank to do the regulation to 5V for you then convert that back down to 3.3V with a simple buck regulator.

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    The maximum charge voltage is specified here as 3.65 Volt.
    – sid_com
    Oct 6 '20 at 14:27
  • @sid_com That's the recommended maximum. They can go up to 4.2V if you want about 1% extra capacity. 3.65 is still above the absolute maximum of an ESP32 anyway.
    – Majenko
    Oct 6 '20 at 14:29
  • 4.2v is typical of lithium chemistries other than lithium IRON phosphate batteries. One can use a radio control type battery charger and set a maximum charge level to lower than the max for the ESP32. Some capacity may be lost in so doing, but with care, it could be accomplished.
    – fred_dot_u
    Oct 6 '20 at 15:15
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    from that page: In the case of LiFePO4 chemistry, the absolute maximum is 4.2V per cell, though it is recommended that you charge to 3.5-3.6V per cell, there is less than 1% extra capacity between 3.5V and 4.2V.
    – fred_dot_u
    Oct 6 '20 at 15:48
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    Exactly. Up to 4.2V. If you have that you will fry the ESP32. You really really should have a battery controller on there to keep it at a regulated 3.3V regardless of the battery voltage (above the lower cutoff voltage of course).
    – Majenko
    Oct 6 '20 at 15:49

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