Why are there no PCI slots in my new computer?
I bουght a nеw computer аnd wanted tο υѕе mу ancient Xonar sound card, bυt tο mу surprise I саn see nο PCI slots. Arе thеу eliminated οr dο thеу јυѕt look differently іn nеw desktop computers? If ѕο, аrе thеrе аnу adapters available?

Well, since you said you BOUGHT a computer, im going to guess its prebuilt and from some huge named company such as Dell, HP, Gateway, etc. These companies use brand name parts for CPU and HDD, but they make PROPRIETARY MOTHERBOARDS. What this means is that they install custom produced mobos for that line of desktop. They are unique and come much less equipped than an aftermarket motherboard from Asus, Gigabyte, etc. Ususally one can only find 1 PCI-E x16 2.0 slot for their graphics option and is rather scarce for any other slots such such as regular PCI, PCI-E x1, and so on. Only high end desktop such as Alienware and stuff along that lone comes with full-fledged custom mobos since they options they offer are limitless. Hope this help, cause i bought a Dell XPS 7100, and all it has is a single PCI-E x16 slot for the ATI 5450 that came with it. It uses the integrated surround channel for sound.
Because it was built on the cheap. No adapters exist.
The simple answer is some modern boards no longer come with PCI, PCIE is the standard to replace PCI/PCI-X/AGP so if you don’t verify a new system (or motherboard) comes with PCI slots, there is a decent chance it wont. This is not limited to OEM vendors, PCI will start completely dissapearing from newer intel chipsets in the future as well and I’m sure AMD will probably follow suit. Unfortunately there is no PCI > PCIE adapter, and I feel your pain, stopped using a decent PCI card in favor of PCIE 1x sound card, luckily there are some excellent inexpensive options for PCIE sound (I believe there is a Xonar pcie 1x that is pretty excellent and can be had for around $40)