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Wanting to run a campaign based on a low-magic, Sword & Sorcery, genre can be difficult with regards to healing. Most class-based, old-school games hang healing solely on the shoulders of a "Cleric" or other priestly class. This includes not only healing generic hit-point damage during an adventure, but also stuff like poisons, diseases, etc. The result of this is that adventuring parties often feel the need to include or hire one or more Clerics as a sort of walking band-aid.
The Cleric as a class concept has its genesis in the "holy" knightly orders of the crusades. It is intended to be a warrior-priest (this is important - clerics weren't necessarily supposed to end up as "divine wizards") with a selection of protective magics. Naturally, the limited number of spell slots and fire-and-forget nature of Vancian casting means most of those "protective magics" get ignored in favor of memorizing the all-important cure spells. It wasn't long before this trend became obvious and Paladins were introduced to occupy the role the Cleric was originally intended to fill.
All that aside, a DM running a campaign steeped in the lore of Leiber, Howard, Vance, and so on, has little use for Clerics
or Paladins from a genre standpoint. The "priests" of the genre are usually more accurately represented by the Magic-User class. But a DM wanting to omit the Cleric altogether from the campaign is basically consigning the players to short lives at worse, and short adventures at best.
Here are some options I've used with my Swords & Wizardry Wilderlands campaign, in which none of the three PCs are clerics.
Basic Healing:
After a combat, damaged PCs may spend one turn on "first aid", healing 1d4hp of damage sustained in that particular combat. This is acually an old house rule I first noticed in the old Judges Guild "Ready Ref" Sheets, and just this house rule alone improves play regardless of genre, as it also frees up Clerics to exploit a wider variety of spells. Personally, I find it also better reflects the nature of hit points themselves, in that they are a more abstract form of damage, representing not just actual wounds, but fatigue, bruising, and even combat ability. The left-over damage after the 1d4hp of first aid represents the actual injuries that will require rest or more intensive healing.
PCs regain lost hp at a rate of 1d3hp per day of rest. This is a healing rule I've adopted from the Moldvay Basic book, and far preferable in my opinion to the more common 1hp/day standard. It's faster on average, and gives the player's something to do every day (roll a d3) rather than just saying "ok, I'm down 6hp, so I rest for 6 days". Also, I consider "day of rest" to be "day spent back in town/camp". No one gets any rest down in the dungeon while nervously anticipating the next wandering Umber Hulk encounter. PCs are welcome to drink, wench, shop, and divvy treasure while "healing", provided there is a comfy bed at the Inn to retire to at a reasonable hour.
PCs regain lost hp at a rate of 2d3 per day under care of a physician. Sometimes, PCs don't have all the time in the world, or are too seriously injured, to play man-about-town. A physician does require his patient to get complete bed rest while under care, and costs roughly 1gp/hp healed. Some adventurous physicians can be lured out on expeditions at a rate of 20-50gp/day (plus cost of poultices used and the standard 1gp/hp healed as a "tip"), but will never venture into actual dungeons, rather remaining at "base camp". Such field physicians sometimes also require the aid of one or more assistants at a rate of 1gp/day.
PCs may be cured of non-magical diseases and poisons under care of a physician. Treatment typically takes 3d4 days and costs roughly 50gp/HD of the creature that inflicted the injury (for instance, disease from a giant rat costs about 25gp to cure). Again, this requires complete bedrest, and a tip of 1gp/hp cured is recommended.
Death and Negative Levels:PCs reduced to 0 or fewer hit points die in 1d6 rounds. This represents the time the PC gets for someone to help stop the bleeding, pull out the arrow, etc, before actually dying. This is totally at the referee's discretion, of course: In cases where the damage inflicted is catastrophic, such as being reduced to -30 hp by a blast of dragon breath, the referee may rule that there is nothing left to heal! This is another discovery from those ever-helpful Ready Ref Sheets.
PCs may regain levels lost as result of supernatural level-draining effects under the care of a qualified physician. Perhaps the most radical of these houserules, its nonetheless a good idea to have something like this in place for any campaign that doesn't have a high-level cleric in every town. Regaining lost levels takes 1 week of complete bedrest for every level lost, and costs 1000gp per level lost. Treatment must begin within one month of the loss, or level loss is permanent. Only physicians with a working knowledge of the supernatural (such as witchcraft, sorcery, religious mysteries, herbalism, etc) may render such treatment.
Non-magical Healing Potions and Poultices:Another good way to supplement the Cleric-free party is to make available some simple herbal or medicinal remedies. Some examples might include:
Poultice of Healing (50gp): This is a creamy, salve-like poultice that is applied directly to a wound. It heals 3-6(1d4+2) points of damage and prevents scarring. Such a poultice is only effective on an individual once every 24 hours. A second application is only half as effective (round fractions down), and further applications have no effect whatsoever, until 24 hours has passed.
Anti-Venom, Weak(25gp): Imbibing this fluid within one round of poisoning allows the poisoned victim an immediate second saving throw against that poison.
Anti-Venom, Strong(200gp): Imbibing this fluid within one round allows the victim an immediate saving throw at +4.
Cleansing Remedy(100gp): Imbibing this foul herbal concoction within 24 hours of contraction of a disease thoroughly and noisily flushes out the victim's system (which may be inconvenient in certain circumstances), giving them an immediate saving throw against the disease with a +2 bonus.
Veritus Charm(500gp): This holy charm, if worn openly around the neck, allows the target of level draining attacks (that wouldn't normally allow a save) a saving throw at -2 against level drain.