Most period properties still have a good deal of Lath and Plaster walls and ceilings. This method of building has long since been superseded, but there are reasons to retain the old as full replacement can be invasive and costly.
We have a number of solutions to repair and retain your lath and plaster even if you want to hang items from your wall.
However, older ceilings that show large or new cracks or have sections coming away require urgent attention and often the best solution is full replacement.
Mouldings and cornicing can normally be left in place with full replacement, simply replacing the heavy failing plaster with lighter, stronger plasterboard.

Before and after lath and plaster replacemnt / repair, retaining repaired cornicing.
Why does lath plaster fail?
Made from a mix of lime mortar, sand and horse hair (other animal hair was used but horses, being the transport of the day, were in ready supply!), traditional plaster relies on "keys" to secure to walls and ceilings. The keys are formed when the plaster is applied to strips of wood called laths. The laths are laid in 1" battons with a 1/8" to 1/4" spacing. Plaster squeezes throught the gaps in the laths forming bulges or keys behind, this then sets and holds the plaster in place. Over time the hair will degrade and the keys fracture, reducing the fixing strength. Eventually, especially on ceilings, the plaster can fail completely and entire sections of ceiling come down. This style of plaster, made essentially from crushed rock is very heavy and able to cause damage to furniture when it fails.
Asbestos and traditional lath and plaster ceilings
Asbestos was not traditionally used in the building industry for the construction of houses with lath and plaster ceilings and walls. Asbestos was a very good component for fireproofing but was not considered beneficial in wall and ceiling construction for domestic dwellings. Furthermore, additional cost and complexity of adding asbestos to a well regarded plaster "mix" would have had your 7-year apprenticeship, time served, plasterer running!
However, during the 1970s and early 1980s, new texturing plasters like Artex, containing gypsum and often asbestos (<4%) were applied on top of traditional plaster ceilings.
Without testing, it is not possible to know if your wall or ceiling plaster contains Asbestos, however it is very unlikely if;
1) There is no additional "skim" coat or additional soft chalky plaster coat on top of the traditional lime mortar. (Lime mortar is gritty as it contains sand, much like cement).
2) The ceiling is smooth. Modern plasters do not contain asbestos and have not done so for 40 years.
Asbestos was added to plaster to give it a firming ability to hold shapes and provide fire proofing (as a secondary benefit). If the ceiling has patterns or shapes, bobbles or swirls and this was applied before 1990 there is a good chance this materials, and only this material DOES contain Asbestos. This can normally be safely removed with steam and a scraper as the steam will significantly reduce friability (the dangerous dust that comes from Asbestos when it is disturbed) However professional assistance where asbestos is likely to be present is recommended.