neofight78, basically, yes. "По" turns adjective in to an adverb. If a word ends on -ому, -ему, -ски, -цки, -ьи then there is a hyphen between the word and "По". It is better to memorise the words you've read or heard from native speakers. For instance: Давай сделаем работу по-быстрому - Let's do the job quickly. But I have never heard "по-медленному" used. It's hard to explain without heaps of rules about grammar and semantics aspects.

"по-настоящему" значит "действительно", "реально" "по-новому" значит "снова", "заново"

neofight78, i believe, Dennis' answer is a good one. basically semantics is : '****** way to do it'. So you may use it with _any_ -ский adjective, including newly invented ones: по-американски, по-женски, по-мужски, по-детски... and also you may say по-быстрому, по-умному, etc. But Dennis has noted a funny thing: people tend not to use it with the qalities considered negative! Still, sometimes I hear 'по-глупому', but much more often you can meet 'по-умному. По -нашему, -вашему, -моему, -твоему, also exist, по-ихнему exist as 'funny' colloquial form (ихний being a dialect form of их. *по-их *по-его *по-её doesn't folow the morphological scheme (they are genitive!), so they sound a way too awkward and doesn't exist). You can easily see from the semantics, why *по-красному, *по-высокому are nonsense. Though the first would be possible if by 'red' we meant a political affiliation, not the colour iself. There also exist 'топить по-белому/по-черному' (heat a bath or a house with hearth or oven. White/black reffered to presence or absence of chimney). Basically you are free to form these phrases as long as they make a clear sense. And where there no interference from other forms. I'd expect 'по-белому' and 'по-черному' applied to manuscripts, as 'черновик' means 'a draft version of a text'. But we have набело(начисто) начерно instead (applicable to any work, not just texts), and even these are largely outdated. Both белый and чёрный are used to describe envy. Черный is used to describe some strong negative feelings. So: завидовать/ненавидеть по-черному, пить по-черному and jokingly завидовать по-белому.

Recently I have seen the phrases по-настоящему & по-новому. 1). How does this construction work and what does it mean? Does the По- prefix plus a dative adjective turn an adjective in to an adverb? What is going on here in terms of grammar and semantics? Я по-настоящему тебя ценю. I really appreciate you. Взглянул на мир по-новому. I looked at the world in a new way. Yes, the prefix по- changes an adjective to an adverb. Настоящий - по-настоящему. Real - really. Новый - по-новому. New - Anew, newly. Братский - по-братскому. Fraternal - fraternally Мужской - по-мужски. Mannish - Mannishly. Дополнительная информация: To clarify: Is there a grammatical rule here, allowing the creation of many of these constructions, or are there a limited number of concrete examples? This condtruction is used only with a limited conctrete examples and there are no such rule for all adjectives.